Topological Function Spaces (Mathematics and its Applications)

Mathematics and Its Applications (Soviet Series)

Managing Editor:

M. HAZEWINKEL Centre for Mathematics and Computer Science, Amsterdam, The Netherlands

Editorial Board: A. A. KIRILLOV, MGU, Moscow, U.S.S.R. Yu. I. MANIN, Steklov Institute of Mathematics, Moscow, U.S.S.R. N. N. MOISEEV, Computing Centre, Academy of Sciences, Moscow, U.S.S.R. S. P. NOVIKOV, Landau Institute of Theoretical Physics, Moscow, U.S.S.R. M. C. POLYVANOV, Steklov Institute of Mathematics, Moscow, U.S.S.R. Yu. A. ROZANOV, Steklov Institute of Mathematics, Moscow, U.S.S.R,

Volume 78

Topological Function Spaces by

A. V. Arkhangel'skii Faculty of Higher Geometry and Topology, Moscow University, Moscow, U.S.S.R.

Kluwer Academic Publishers Dordrecht / Boston / London

Library of Congress Cataloging-in-Publication Data available from the Publisher.

ISBN 0-7923-1531-6

Published by Kluwer Academic Publishers, P.O. Box 17, 3300 AA Dordrecht, The Netherlands. Kluwer Academic Publishers incorporates the publishing programmes of D. Reidel, Martinus Nijhoff, Dr W. Junk and MTP Press. Sold and distributed in the U.S.A. and Canada by Kluwer Academic Publishers, 101 Philip Drive, Norwell, MA 02061, U.S.A. In all other countries, sold and distributed by Kluwer Academic Publishers Group, P.O. Box 322, 3300 AH Dordrecht, The Netherlands.

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All Rights Reserved 0 1992 Kluwer Academic Publishers No part of the material protected by this copyright notice may be reproduced or utilized in any form or by any means, electronic or mechanical, including photocopying, recording or by any information storage and retrieval system, without written permission from the copyright owner. Printed in the Netherlands

SERIES EDITOR'S PREFACE

'Et moi, .... si j'avait su comment an revetir, je n'y serais point alle:' Jules Verne

The series is divergent; therefore we may be able to do something with it. 0. lleaviside

One service mathematics has rendered the human race. It has put common sense back

where it belongs, on the topmost shelf next to the dusty canister labelled 'discarded nonsense. Eric T. Bell

Mathematics is a tool for thought. A highly necessary tool in a world where both feedback and nonlinearities abound. Similarly, all kinds of parts of mathematics serve as tools for other parts and for other sciences. Applying a simple rewriting rule to the quote on the right above one finds such statements as: 'One service topology has rendered mathematical physics ...'; 'One service logic has rendered computer science ...'; 'One service category theory has rendered mathematics ...'. All arguably true. And all statements obtainable this way form part of the raison d elrc of this series.

This series, Mathematics and Its Applications, started in 1977. Now that over one hundred volumes have appeared it seems opportune to reexamine its scope. At the time I wrote

"Growing specialization and diversification have brought a host of monographs and textbooks on increasingly specialized topics. However, the 'tree' of knowledge of mathematics and related fields does not grow only by putting forth new branches. It also happens, quite often in fact, that branches which were thought to be completely disparate are suddenly seen to be related. Further, the kind and level of sophistication of mathematics applied in various sciences has changed drastically in recent years: measure theory is used (non-trivially) in regional and theoretical economics; algebraic geometry interacts with physics; the Minkowsky lemma, coding theory and the structure

of water meet one another in packing and covering theory; quantum fields, crystal defects and mathematical programming profit from homotopy theory; Lie algebras are relevant to filtering; and prediction and electrical engineering can use Stein spaces. And

in addition to this there are such new emerging subdisciplines as 'experimental mathematics', `CFD', 'completely integrable systems', 'chaos, synergetics and large-scale order', which are almost impossible to fit into the existing classification schemes. They draw upon widely different sections of mathematics."

By and large, all this still applies today. It is still true that at first sight mathematics seems rather fragmented and that to find, see, and exploit the deeper underlying interrelations more effort is needed and so are books that can help mathematicians and scientists do so. Accordingly MIA will continue to try to make such books available. If anything, the description I gave in 1977 is now an understatement. To the examples of interaction areas one should add string theory where Riemann surfaces, algebraic geometry, modular functions, knots, quantum field theory, Kac-Moody algebras, monstrous moonshine (and more) all come together. And to the examples of things which can be usefully applied let me add the topic 'finite geometry; a combination of words which sounds like it might not even exist, let alone be applicable. And yet it is being applied: to statistics via designs, to radar/ sonar detection arrays (via finite projective planes), and to bus connections of VLSI chips (via difference sets). There seems to be no part of (so-called pure) mathematics that is not in immediate danger of being applied. And, accordingly, the applied mathematician needs to be aware of much more. Besides analysis and numerics, the traditional workhorses, he may need all kinds of combinatorics, algebra, probability, and so on. In addition, the applied scientist needs to cope increasingly with the nonlinear world and the

vi

SERIES EDITOR'S PREFACE

extra mathematical sophistication that this requires. For that is where the rewards are. Linear models are honest and a bit sad and depressing: proportional efforts and results. It is in the nonlinear world that infinitesimal inputs may result in macroscopic outputs (or vice versa). To appreciate what I am hinting at: if electronics were linear we would have no fun with transistors and computers; we would have no TV; in fact you would not be reading these lines. There is also no safety in ignoring such outlandish things as nonstandard analysis, superspace and anticommuting integration, p-adic and ultrametric space. All three have applications in both electrical engineering and physics. Once, complex numbers were equally outlandish, but they frequently proved the shortest path between 'real' results. Similarly, the first two topics named have

already provided a number of 'wormhole' paths. There is no telling where all this is leading fortunately. Thus the original scope of the series, which for various (sound) reasons now comprises five sub-

series: white (Japan), yellow (China), red (USSR), blue (Eastern Europe), and green (everything else), still applies. It has been enlarged a bit to include books treating of the tools from one subdiscipline which are used in others. Thus the series still aims at books dealing with:

a central concept which plays an important role in several different mathematical and/or scientific specialization areas; new applications of the results and ideas from one area of scientific endeavour into another;

influences which the results, problems and concepts of one field of enquiry have, and have had, on the development of another.

Let X be a topological space, possibly with extra structure. One can stay at this level to study things. But just as in the case of the physical world around us, one should not be satisfied with what

can be perceived by the unaided eye. One likes to use microscopes and high energy particle accelerators, and indeed without these tools our understanding would have been very limited. There are also mathematical microscopes, and, in the case at hand, topology, one of these microscopes, is the space of functions on a topological space. An early instance (historically speaking) is Fourier analysis, which is invisible at the level of the circle acting on itself (which is an irreducible object), but which certainly is a most powerful tool at the level of functions. It is quite amazing what can be understood and perceived of a space at the level of functions on it. At the time, the famous book of Gillman and Jerison was an eye opener (except to the initiates); since then much more has happened. Those who doubt the power of function spaces as an aid to understanding need only peruse the present volume. It should suffice to convince even the most sceptical. The shortest path between two truths in the

real domain passes through the complex domain.

J. Hadamard La physique ne noon donne pas seulement )'occasion de resoudre des probl3nes ... eae nous fait pressentir la solution. H. Poincare

Bussum, September 1991

Never lend books, for no one ever returns than; the only books I have in my library are books that other folk have lent me. Anatole France

The function of an expert is not to be more right than other people, but to be wrong for more sophisticated reasons.

David Butler

Michiel Hazewinkel

Contents

Series Editor's Preface

Chapter 0. General information on Cp(X) as an object of topological algebra. Introductory material 1. General questions about Cp(X) 2. Certain notions from general topology. Terminology and notation 3. Simplest properties of the spaces CP(X,Y) 4. Restriction map and duality map 5. Canonical evaluation map of a space X in the space CCp(X) 6. Nagata's theorem and Okunev's theorem

v

1 1

4 9 11

16 22

Chapter I. Topological properties of Cp(X) and simplest duality theo25 1.

Elementary duality theorems

When is the space CC(X) o,-compact? 3. Cech completeness and the Baire property in spaces CC(X) 4. The Lindelof number of a space Cp(X), and Asanov's theorem 5. Normality, collectionwise normality, paracompactness, and the extent of Cp(X) 6. The behavior of normality under the restriction map between function spaces 2.

25

28 31

33 36

43

Chapter II. Duality between invariants of Lindelof number and tightness type 45 1. Lindelof number and tightness: the Arkhangel'skii-Pytkeev theorem 2.

45

Hurewicz spaces and fan tightness

48

viii

CONTENTS

3.

Frechet-Urysohn property, sequentiality, and the k-property of

CC(X)

4. 5.

6. 7. 8.

51

Hewitt-Nachbin spaces and functional tightness Hereditary separability, spread, and hereditary Lindelof number Monolithic and stable spaces in Cp duality Strong monolithicity and simplicity Discreteness is a supertopological property

57 66 76

83

87

Chapter III. Topological properties of function spaces over arbitrary compacts 91 1. Tightness type properties of spaces CC(X), where X is a compactum, and embedding in such CC(X) 91 2.

Okunev's theorem on the preservation of o-compactness under t-

equivalence

97

Compact sets of functions in C,(X). Their simplest topological properties 102 4. Grothendieck's theorem and its generalizations 106 5. Namioka's theorem, and Pt'ak's approach 115 6. Baturov's theorem on the Lindelof number of function spaces over compacta 121 3.

Chapter IV. Lindelof number type properties for function spaces over compacta similar to Eberlein compacta, and properties 125 of such compacta 1. Separating families of functions, and functionally perfect spaces 125 2. Separating families of functions on compacta and the Lindelof number of C,,(X) 3.

131

Characterization of Corson compacta by properties of the space

CD(X)

136

4. Resoluble compacta, and condensations of C,(X) into a E.-product

of real lines. Two characterizations of Eberlein compacta

144

The Preiss-Simon theorem 152 6. Adequate families of sets: a method for constructing Corson compacta 156 5.

7.

The Lindelof number of the space Cp(X ), and scattered compacta 164

CONTENTS

The Lindelof number of CP(X) and Martin's axiom 9. Lindelof E-spaces, and properties of the spaces Cy,,,(X) 8.

ix

168 174

10. The Lindelof number of a function space over a linearly ordered compactum 181 11. The cardinality of Lindelof subspaces of function spaces over compacta 185 Bibliography Index

193

203

CHAPTER 0

General information on C,,(X) as an object of topological algebra. Introductory material

1. General questions about C,(X) The basic ohjrcl. of this hook is the space (,,(X) of all realvalucd continuous fou rtions on a topological space X in the topology of pointwise convergence. The space C,,(X) deserves attention in many respects. First, the transition from X to CC(X) is of interest; it can be naturally considered as a fundamental operation over X. If X is discrete, this operation becomes the operation of taking a power: in

this case CC(X) coincides with R. If X is a 'standard' space (an interval, the line, Hilbert space, the Tikhonov cube, etc.), the corresponding space Cp(X) can also be naturally understood as a standard object, and it is appropriate to compare it with other topological spaces. As we will see, in its properties the space CC(X) can be very different from the space

X. This gives a basis for the expectation that CG(X) will be useful in constructing examples. Finally there arises a fundamental question: how are the properties of X and Cc(X) related? Special interest is attached to `duality' properties of X and- CC(X), i.e.

properties characterizing each other. In considering this question we see that X and CC(X) are not on the same foot: on X there is only a topological structure, while Cp(X) carries at the same time a topology and two natural algebraic operations of addition and multiplication, making it a topological ring. This allows us to regard Cp(X), depending on the purpose, as a topological space, a topological ring, a topological group, or a linear topological space, opening up the possibility of `sorting' the properties of X in relation to whether they are determined by the algebraic structure of the ring Cp(X), depend on the properties of CC(X) as a linear topological space, or can be fully characterized by purely topological properties of CG(X).

In relation to problems in functional analysis and problems in general topology itself, it is useful to know which compacta can be realized as subspaces of function spaces. In this way are distinguished, e.g., Eberlein co?npacta (i.e. compact subsets of Banach spaces in the weak topology). The weaker the topology on the function space (i.e. the smaller the amount of open i

2

0. GENERAL INFORMATION ON Cp(X). INTRODUCTORY MATERIAL

sets in it), the larger will be the amount of compacta in this space. One of the greatest merits of the topology of pointwise convergence is the fact that it is the smallest of

practically all natural topologies on a function space, and hence gives the largest amount of compacta. However, the main advantage of the topology of pointwise convergence over the compact-open topology, the topology of uniform convergence, and other topologies is expressed in the following remarkable theorem of Ju. Nagata [124]: Two Tikhonov spaces X and Y are homeomorphic if (and only if) the topological rings CC(X) and CC(Y) are (topologically) isomorphic. Thus, the topological ring CC(X) carries complete information on the properties of X. At the same time, the ring CC(X) is always algebraically isomorphic to the ring Cp(v X ), where vX is the Hewitt-Nachbin compactification of X (the isomorphism is realized by extending an f E CC(X) to f E C9(vX)). Hence, the algebraic properties of the ring Cp(X) only are not sufficient to `detect' properties of X such as compactness and functional completeness. Now we can state some typical problems related with Cp(X), as follows. General problem A. Establish a correspondence between properties of the topological space X and of the topological ring CC(X). The following problems are distinguished within this general problem. 1. Distinguish the properties of X that are characterized by the topological properties of CC(X) only. Such properties of X are naturally called supertopological. Are there general tests for being a supertopological property? 2. Distinguish the properties of X corresponding to the properties of the linear topological space Cp(X). Such properties of X are conveniently called linear topological. Is there a general `intrinsic' test for such properties? 3. Find the properties of X that are characterized by the algebraic properties of the ring C9(X) only (algebraic-topological properties). How general are these properties (e.g., from the category point of view)? 4. Which properties of X depend really on the properties of CC(X) as a topological ring? Such properties are naturally called ring properties. Are there any means to guess in advance that a property is a ring property?

However, Cp(X) can be regarded also as a topological group, and also as a uniform space. In relation to this, analogs of the problems 1-4 can be properly formulated.

5. Which properties of X are characterized (and in what way) by the properties of CC(X) as a uniform space? Such properties of X are called u-properties. The general problem A aims at the discovery of duality properties. However, the questions 1-5 allow a more general treatment. General problem A. Let Cp(X) and Cp(Y) be identical in some sense, as topological spaces, as linear topological spaces, or as algebraic rings. Which properties will then be common to the spaces X and Y? This statement of the question does not require that we must find the property dual to a property P of X whose invariance has been proved. However, the presence

I. GENERAL QUESTIONS ABOUT Cp(X)

3

of the dual property of P guarantees, of course, the invariance of P under the isomorphism from Cp(X) to Cp(Y) which preserves the structure in terms of which this dual property is formulated. Thus, the following four problems arise within problem A. 1'. Let Cc(X) and Cp(Y) he homeomorphic. Which are the properties that the spaces X and Y can have simultaneously only? Of course, here we are interested not only in `positive' results, but also in examples showing how far away from each other the properties of X and Y can be if CC(X) and Cp(Y) are homeomorphic. 2'. Suppose the linear topological spaces CC(X) and Cp(Y) are linearly homeomorphic. How are the properties of the topological spaces X and Y related in this case? 3'. Which properties can be transferred from X to Y if the rings Cp(X) and cp(Y) are algebraically isomorphic? 4'. Suppose the uniform spaces (.I,(X) and 0p(1') are nnifornlly homavnuorlahic. Which are the properties that, the topological spaces X and Y can have sinntlt.aneously only?

One general approach to the classification of topological spaces is the study of the possible homeomorphic eanbeddings of the spaces of one class into spaces of another class. This is related to the following General problem B. Suppose we are given a class P of topological spaces. Char-

acterize in an 'intrinsic' way the class f(P) of all spaces Y for which there is an X E P such that Y is homeomorphic to a subspace of Cc(X). Finally, the general problems A, A', and B do not encompass all interesting questions related to CC(X). They only represent some important directions in the study of function spaces within general topology and topological algebra. In particular, it is expedient to study CC(X) by itself, without directly relating it to the initial spaces X. It is to be expected that the peculiarities of the structure of spaces Cc(X) would allow us to prove specific theorems about the relations between topological properties of such spaces. E.g., it is easy to prove that Cp(X) is paracompact if and only if it is Lindelof. Note that the presence of one sufficiently rich algebraic structure (e.g., that of a group) compatible with the topology can considerably 'improve' the topological properties, and may change the relations between these in a fundamental way. E.g., for any topological group metrizability is equivalent to the first axiom of countability. In this direction of research the following unsolved problems deserve attention.

Group of problems C. 1. Let Cp(X) he Lindelof. Is it true that CC(X) x Cp(X) is a Lindelof space? 2. Let CC(X) be normal. Is it true that CC(X) x CC(X) is normal? 3. Can the space Cc(X) always be continuously mapped onto the space Cc(X) x Cp(X)? S. P. Gul'ko has proved that the space Cc(T(c'1 -h 1)) is not homeomorphic to its square, see also W. Marciszewski [113].

4

0. GENERAL INFORMATION ON Cp(X). INTRODUCTORY MATERIAL

2. Certain notions from general topology. Terminology and notation In notation and terminology we follow almost without exception R. Engelking's book [661. The spaces' considered in this book are taken to be Tikhonov spaces (i.e. completely regular spaces in which finite sets are closed), provided no precise assumptions concerning separation axioms are made. The symbols X, Y, Z, T (with accents or indices) are used only for topological spaces; the notation Y C X means that Y is a subspace of the space X. The closure of a set A C X in X is denoted by A or cl X (A). Further, R denotes

the ordinary space of reals, N+ = [1,2.... }, N = {0} U N+, I = [0,11 is the unit interval with the ordinary topology, and IT is the Tikhonov cube of weight r. The cardinality of a set A is denoted by 1AI. A set A is called countable if (AI < No, where No is the smallest infinite cardinal. Further, D = {0, 1) is the discrete two-point space, D, is the discrete space of cardinality r, and A, denotes the one-point compactification (in the sense of P. S. Aleksandrov, by the point a,) of the discrete space of cardinality T. A similar space L, is obtained by adjoining to D, a new point p, and declaring the neighborhoods of p, to be those sets containing p, whose complements are countable (the points of D, remain isolated). This space L, is called the one-point Lindelofccation of the discrete space of cardinality r. By X+ we denote the space obtained by adjoining one new isolated point to the space X.

If r is a cardinal, then r+ denotes the first cardinal larger than r. If a is an ordinal, than a + 1 is the first ordinal larger than a. As usual, No is the smallest infinite cardinal and xI = (No)+ is the first uncountable cardinal. By WI we denote the smallest ordinal of cardinality l't1i and the cardinal No will often be identified with the smallest countable ordinal w (the order type of the set N+, <).

ExpX denotes the set of all subsets of a set X, and XA denotes the characteristic function of a set A C X. The symbols r, A usually denote infinite cardinals only. A well ordered set W, < is called minimal if 1{x E W: x < y}J < IWI for all y E W. The cofinality cf(r) of a cardinal r is the smallest cardinal X such that the set of cardinality r can be represented as the union of some family, of cardinality at most A, of sets of cardinality less than T. A map f : X Y is called a map onto if f (X) = Y. A condensation is a bijective continuous map onto. An important place in this book is occupied by cardinalvalued (or cardinal) topological invariants. Here follows a short list of the most important among these. The Lindelof number 1(x) of a space X is the smallest infinite cardinal r such that any open cover of X contains a subcover of cardinality < r.

The Suslin number c(X) of X is the smallest infinite cardinal r such that the cardinality of every family of pairwise disjoint nonempty open sets in X does not exceed r. The weight w(X) of X is the minimal cardinality of a base of X. This invariant has a generalization which is very useful in C, -theory. Recall that a network in a space X is a family S of subsets of the set X such that for any point x E X and any

2. CERTAIN NOTIONS FROM GENERAL TOPOLOGY. TERMINOLOGY AND NOTATION

5

neighborhood Ox of x there is a P E S such that x E P C Ox. The network weight nw(X) of X is the minimal cardinality of a network in X. The density d(X) of X is the minimal cardinality of an everywhere dense set in X. Thespread s(X) of X is the smallest infinite cardinal r such that the cardinality of every discrete subspace of X does not exceed T. The extent e(X) of X is the smallest infinite cardinal r such that the cardinality of every closed discrete subspace of X does not exceed T. iw(X) denotes the minimal weight of all spaces onto which X can be condensed. The cardinal invariant iw(X) is called the i-weight of X. These cardinal invariants have a global character: they describe properties of the

space as a whole. Not less important are 'point' cardinal invariants, containing information on the structure of the space at a given point. of it, and on peculiarities of limit transition in this space. Here are the definitions of some of them. The tightness t(X) of X is the smallest infinite cardinal r such that for any set

A C X and any point xEX there is a set B C X for which CBI
set A there is a set U E ry such that A C U C OA, and if also A C fly, then -y is called a base of X at A. The minimal cardinality of all such families ry is called the character of X at A, and is denoted by X(A,X). If A = {x} is a singleton, then instead of X({x}, X) and ?'({x}, X) we write X(x, X) and t/i(x, X), respectively, and call them the character and pseudocharacter of X at. X.

The diagonal of a space X in its square is the set 0x = {(.T., x): x E X), also simply denoted by A. The diagonal number L(X) of X is the pseudocharacter of its

square X x X at its diagonal, i.e. 0(X) _ ti(Ox,X x X). On the class of Tikhonov spaces we consider only those cardinalvalued functions whose values do not change tinder transition to a homeomorphic space, 'i.e. the cardinalvalued topological invariants. It is also convenient to assume that in all definitions of cardinalvalued invariants, including those given above, we have in mind infinite cardinals only. Thus, the smallest value that a cardinal invariant can take is Ho. If ¢ is a cardinal invariant, then h¢ denotes the new cardinal invariant defined as follows:

hO(X) = sup{¢(Y) : Y C X}; here X is an arbitrary topological space. We also put

¢'(X) = sup{¢(X') : n E N+}. There is a number of other methods for constructing new cardinal invariants from a given invariant 0, but the two methods given above are very important for us. A space is called o-compact (o-countably compact, etc.) if it is the union of a countable set of compact (respectively, countably compact, etc.) subspaces.

0. GENERAL INFORMATION ON Cp(X). INTRODUCTORY MATERIAL

6

A space is called pseudocompact if every realvalued continuous function on it is bounded. A subset A of a space X is called bounded or R-bounded (in X) if every realvalued continuous function on X is bounded in absolute value on A.

A space X is called a p-space if the closure of each set that is bounded in X is compact.

A subspace A of a space X is C-embedded (C°-embedded) in X if every realvalued continuous (bounded continuous) function on A can be extended to a realvalued R can be extended to a continuous function on all of X. If every function f : A realvalued continuous function on all of X, then A is said to be R-embedded in X. For two arbitrary functions fl, f2 E Cp(X) the functions h = min{ fi, f2} and g = max{ fl,

f2} are naturally defined: h(x) = min{fi(x), f2(x)} and g(x) = max{fi(x), f2(x)}

for allxEX. A set A C X is called a zero set (or Z-set) if there is on X a realvalued continuous function f such that f '(0) = A. The complement of a zero set in X is called a (-.ozero set. Certain classes of topological spaces will be regarded very often in this book. We recall the definitions of some of these. A space X is called a k-space if in it every set for which the intersection with an arbitrary compactum in X is closed in this compactum, is closed. If each point x E X is contained in a compactum 4) C X of countable character in X, then we say that X is a space of pointwise countable type. The class of Lindelof E-spaces is defined as the smallest class of spaces containing all compacta, all spaces with a countable base, and closed tinder the following three operations: taking the product of two spaces, transition to a closed subspace, and transition to a continuous image. It is well known 11231 that a space Z is a Lindelof E-space if and only if it can be represented as a continuous image of a space Y which can be perfectly mapped onto a space with a countable base. Here, a map is called perfect if it is continuous, closed, and pre-images of points are compact. A space is called a P-space if the intersection of any countable family of open sets in it is open. Spaces homeomorphic to closed subspaces of some power R' of the reals are called Hewitt-Nachbin complete or R-complete. A space is called Dieudonne complete if it is complete relative to the maximal uniform structure compatible with its topology.

A cardinal T is called Ulam measurable if on the set of cardinality T there is a maximal centered system with empty intersection, such that the intersection of any countable subfamily of it is not empty (i.e. countably centered). In consistency with the axioms of set theory of the system ZFC (of ZermeloFraenkel), we may assume that measurable cardinals do not exist. Under this assumption every Dieudonne complete space is Hewitt-Nachbin complete. The converse is always true.

2. CERTAIN NOTIONS FROM GENERAL TOPOLOGY. TERMINOLOGY AND NOTATION

7

A space X has the Baire property if in it the intersection of any countable family of open everywhere dense sets is everywhere dense.

A space is called Cech complete if it can be represented as the intersection of a countable family of open sets in some ambient compactum. Each such space has the Baire property. A metrizable space is Cech complete if and only if it is metrizable by a complete metric. A space is called scattered if each nonempty subspace Y of it contains an isolated (in Y) point. A space X is called sequential if each nonclosed set A C X contains a sequence of points {xn: n E N+} that converges to some point x E X \ A. If for each set A C X and each point x E A there is in A a sequence {x,,: n E N+} converging to x, then X is called a Frechet-Urysohn space. Continuous images of the space of irrational numbers are called analytic spaces. A space X is of type Ko6 if it is the intersection of a countable family of o-compact subspaces of some ambient space. Continuous images of spaces of type Ks arc called K-analytic spaces.

Let X = fl{Xa: a E Al be a topological product and x' = {x.*: a E Al a point of it. Then the E-product of this spaces Xa with respect to a E A and with base at the point x' is the subspace E jj{Xa: a E A} = {x E X: ]{a E A:xa # xQ}l < lio} of the product f[{Xa: a E Al formed by all points that differ from x' on at most a countable set (of coordinates). We also consider the a-product u fj{Xa: a E A); it consists of all points x E II{Xa: a E A} whose coordinates differ from the coordinates of x' only for finitely many coordinates a E A. The following assertion can be proved by standard methods [16], [66].

0.2.1. Proposition. If Y is an everywhere dense subspace of a product X = fI{,Ya: a E A} and 45 C Y is a nonempty compactum of countable character in Y (i.e. X(4', Y) < lio), then the set of all a E A for which Xa is not compact is at most countable.

We will also need the following assertion [13], [16].

0.2.2. Proposition. If Ia == I = [0, 1] for a E A and Y is a subspace of the Tikhonov cube IA = fj{Ia: a E Al which, whatever the countable set B C A, projects under the canonical projection Ire: 1A IB onto the whole cube IB = fj{Ia: a E B} of IA, then Y is pseudoeompact.

Finally, an important part is played by the

0.2.3. Factorization lemma [13], [66].

Let X = n{Xa: a E A) be a product

of spaces with a countable base, Y an everywhere dense subspace, and f : Y --> R a continuous function on Y. Then there are a countable subset B C A and a realvalued continuous function g on the subspace pB(Y) of the space fj{ Xa: a E B} which is the

0. GENERAL INFORMATION ON Cp(X). INTRODUCTORY MATERIAL

8

image of Y under the canonical projection pB of the space fl{Xa: a E A} onto the space fl{Xa: a E B}, such that g(pB(y)) = f(y) for ally E Y, i.e. f = g o pB. This lemma can be proved similarly to the case when Y = X [16]. Many more general factorization theorems for continuous functions on everywhere dense subspaces of products can be found in [13]. By C(X,Y) we denote the set of all continuous maps from a space X into a space

Y. We set C(X) = C(X,R) and C°(X) = If E C(X): f is bounded}. By Cp(X) and Cp°(X) we denote, respectively, the sets C(X) and C°(X) endowed with the topology of pointwise convergence. The topological product RX of X copies of the real line R

is interpreted as the set of all maps from X into R, endowed with the topology of pointwise convergence.

If AcX andBcY,then (A,B)={f EC(X,Y): f(A)cB}. Let £ be a family of sets in X, with 0 E E. Then the family Pe of all sets of the form (A, U), where A E £ and U is an open set in Y, is a subbase for some topology Ye on the set C(X, Y); it is called the topology of uniform convergencc on the elements of the set E. If £ is the family of all finite subsets of X, then TE is called the topology of pointwise convergence; endowed with this topology, C(X, Y) will be denoted by Cp(X,Y). If £ is the family of all compact subsets of X, then Te is called the compact-open topology; endowed with this topology, C(X, Y) will be denoted by CC(X,Y). If £ is the family of all subsets bounded in X, then C(X,Y) endowed with Ye will be denoted by C°(X, Y).

The standard base of the space Cp(X, Y) consists of the sets W (xl, ... ) xk, , Uk) = If E C(X, Y): f (xi) E U{, i = 1, ... , k}, where XI.... , xk E X)

UI,

Ul,... , Uk are open sets in Y, and k E N+. The spaces CC,n(X), where n E N+, are defined by induction as follows: Cp,l(X) _ Cp(X), and Cp,n+I(X) = Cp(CC,n(X)). The space C,,2(X) will also be denoted by CPCP(X).

Two spaces X and Y are called t-equivalent (l-equivalent, u-equivalent), written

as X L Y (respectively, X L Y, X u, Y), if the spaces Cp(X) and Cp(Y) are homeomorphic (respectively, linearly homeomorphic, uniformly homeomorphic). If X is homeomorphic to Y we write X n Y. Clearly,

X-Y=X_Yr+XLY. These implications cannot be reversed.

0.2.4. Problem. Find a construction that allows one to obtain spaces X and Y such that X £ Y but X not 1-equivalent to Y. If Cp,n(X) and Cp,n(Y) are homeomorphic (linearly homeomorphic, uniformly home-

omorphic), we write X _'" Y (respectively, X II" Y, X u Y). It is obvious that XI"X`"=' Y.

3. SIMPLEST PDOPERTIES OF THE SPACES Cp(X,Y)

9

3. Simplest properties of the spaces CP(X, Y) The following elementary properties of the spaces C,,(X,Y) can he proved without difficulty.

0.3.1. Proposition. Let 13 be a base of a space Y. Then {W (xl,... , xk, U) , ...

, Uk):

xi E X, U1 E 8, i = 1, ... , k, k E N+} is a base of the space C,(X, Y).

0.3.2. Proposition. If Y C Z, then CC(X, Y) is a subspace of the space Cp(X, Z); it is closed if Y is closed in Z.

0.3.3. Proposition. The space Cp(X, H{Y,,: a E A}) is canonically homeomorphic to the space f [{cp(X,Y.): a E A;.

0.3.4. Proposition.

Let X = E®{X,,: a E A} be the free topological sum of the spaces Xa. Then for any space Y the space Cp(X',Y) is canonically homeomorphic to the space f({CP(X(,,Y): a E A}.

In the sequel our main attention will be given to the space Cp(X). For k E N+,

XI,...,xkE X, f EC(X),andE>0we put W (f, x1, ... , xk, E) = {9 E C(X) : l9(xi) - f (xi)I < E, i = 1, ... , k}. xk, e) is a base of Cp(X ). Clearly, the family of all sets of the form W = W (f, iI, ,...

0.3.5. Proposition. C'(X) is a locally convex [631 linear topological space over the field R.

0.3.6. Proposition.

The space Cc(X) is a subspace of the space RX; moreover, Cp(X) = RX, i.e. Cp(X) is everywhere dense in RX.

Proof. The definition of the topology of the product RX and of the topology of pointwise convergence in Cp(X) imply that Cp(X) is a subspace of RX. Let f E Rx.

F o r any finite collection X1,... ,xk E X there is a function g E Cp(X) such that 9(xi) = f(xi), i = 1,... ,k. Hence f E Cc(X).

0.3.7. Corollary. The Suslin number of a space Cp(X) is always countable. This follows from the facts than, c(RX) < 1 o 116, Chapt. 2, No. 3831 and Cp(X) _ RX.

Assertion 3.7 is remarkable for its combination of generality and nontriviality: it expresses something highly nontrivial about the topological structure of an arbitrary Cp(X). It implies, e.g., that for an arbitrary space of the form Cp(X) paracompactness and the Lindelof property are equivalent (cf. Chapt. 1, §5). Moreover, since every Dieudonne complete space with countable Suslin number is Hewitt-Nachbin complete 1161, 1661, 3.7 implies

0. GENERAL INFORMATION ON Cp(X). INTRODUCTORY MATERIAL

10

0.3.8. Theorem.

The space Cp(X) is Kewitt-Nachbin complete if and only if it

is Dieudonne complete.

Under a continuous map the Suslin number does not increase. Hence, if Cp(X) can be continuously mapped onto X, then c(X) < c(Cp(X)) < Ro. Assertion 3.7 can be strengthened considerably. A cardinal r is called a precaliber (caliber) of a space X if, for any family µ = {lln: a E A} of nonempty open sets in X such that CAI = r, there is a B C A for which IBI = r and the family {Ua: a E B} is centered (respectively, CBI = r alid n{U,,:

a E B} 0 0). The following four useful assertions about calibers and precalibers can be proved rather simply.

0.3.9. Proposition. Let Y be an everywhere dense subspace of a space X. Then a cardinal T is a precaliber of Y if and only if it is a precaliber of X.

0.3.10. Proposition.

If r is a regular cardinal with r > d(X) (where d(X) is the density of X), then r is a caliber of X.

0.3.11. Proposition.

If r is a precaliber of a space X, the cardinality of every disjoint system of nonempty open sets in X is strictly less than r.

0.3.12. Proposition. If X is a compactum, then T is a precaliber of X if and only if r is a caliber of X. The following classical result is due to N. A. Shanin [3], [16].

0.3.13. Theorem. If a regular cardinal r is a precaliber (caliber) of a space X., for all a E A, then r is also a precaliber (respectively, a caliber) of the product X = Fl{X,:

aEA}. Since the space R is separable, assertions 3.6, 3.9, 3.10, and 3.13 imply

0.3.14. Corollary. Every uncountable regular cardinal is a precaliber of Cp(X). The following assertion gives yet another nonobvious peculiarity of the topological structure of every space CC(X).

0.3.15. Proposition.

The closure of any open set in Cp(X) is the zero set of some continuous function g: CC(X) R (i.e. is a Z-set).

Proof. a. Let U be an open set in RX. We show that U is the zero set of some realvalued continuous function on Rx. By Zorn's lemma [66] there is a maximal disjoint system -y of standard open sets

4. RESTRICTION MAP AND DUALITY MAP

in RX contained in U. For each V = W (f , xl, ..

, xk, e)

11

E -y we put k(V) =

{x1,... ,xk}. The maximality of -y implies that 07,y = U. But c(RX) < Ro (cf. 3.6). Hence the family -y is countable. Therefore the family L = U{k(V): V E y} is countable. Consider the projection ir: RX --: RL (where ir(f) = AL for all f E RY). Clearl , it is a continuous open map from RX onto RL, and Uy = ir-'(ir(Uy)). Hence

-'(ir(uy)) =U = U. Since the weight of RL is countable, there is a realvalued continuous function g

on RL for which g-'(0) = ir(Uy). Put f = g o it. Then f E C(RY) and f-'(0) _ 7_'(g-'(0)) = U, i.e. U is the zero set of f. b. We show that if Y C X, Y = X, and in X the closure of each open set is the zero set of some continuous function f : X - R, then the space Y has the same properties.

Let U C Y and U open in Y. Take an opensetUinXsuchthatUflY=U. The requirement Y = X implies that clx(U) = clx(U). There is (by the assumption) a function f E C(X) such that clx(U) = f-'(0). For the function g = fly E Cr,(Y) we then have g-1(0) = f-'(0) fl Y = clx(U) fl Y = cly(U). Assertion 3.15 now follows from a), b), and the fact. that c (X) is everywhere dense in Rx (see 3.6). E. V. Shchepin calls a space X perfectly-ic-normal 165] if the closure of each open set.

in it is the zero set of some realvalued continuous function on X. In this terminology, proposition 3.15 can-be restated as follows:

0.3.16. Proposition. The space CC(X) is always perfectly-x-normal.

4. Restriction map and duality map Let Y be a subspace of a space X. By it = iry: Cp(X) -+ Cp(Y) we then denote the map of restricting a function in C9(X) to Y, i.e. iry(f) = f by for all f E Cp(X). The subspace iry(Cp(X)) C Cp(Y) is denoted by Cp(YIX).

0.4.1. Proposition. For any Y C X the following hold: 1) the map it is continuous and 7r(Cp(X)) = Cp(Y);

2) if Y is closed in X, then it is an open map from Cp(X) onto the subspace 'r(Cp(X)) of Cp(Y); 3) if Y is compact, then ir(CC(X)) = Cp(Y);

4) If X is normal and Y is closed in X, then ir(Cp(X')) = Cp(Y); 5) if Y is everywhere dense in X, then 7r: Cp(X) 7r(Cp(X)) is a hijective continuous map, i.e. a condensation. Proof. 1. it is clearly continuous. We prove that ir(Cp(X)) = Cc(Y). Take arbitrary g E Cp(Y), e > 0, yl,... , yk E Y. Since X is a Tikhonov space, there is a function f E Cp(X) for which f (yi) = g(yi), i = 1,... , k. Then ir(f) E W (g, y1i ... , ilk, e). 2. Consider an arbitrary 14"(f, xl, ... , xk, e) c Cp(X ). We may assume that

x1,... xl E Y and xi+1, x.e E X \ Y, 0 < l < k. Clearly W(f,xl,... ,Xk,f) C W(ir(f),x1,... ,xl,e) n 7r(Cp(X)). We show that 1r(W(f,xl,... ,Xk,C)) = n7r(Cp(X)), which implies that the set W(ir(f),x1,... ,xi,e)

0. GENERAL INFORMATION ON Cp(X). INTRODUCTORY MATERIAL

12

is open in 7r(Cp(X)); in turn, this means that the map ir: C,,(X) -> ir(CC(X)) is open.

Let g E ir(Cp(X)) and Jg(xi)-ir(f)(xi)J < e, i = 1,... l. Fix g, E CC(X) such that 7r(g1) = g. Since X is completely regular, Y is closed in X, and xl+I,... , xk E X \ Y,

there is a function 0 E C,,(X) such that O(Y) = {0} and O(x5) = f(xj) - g,(xj), j = 1 + 1,... , k. Put h = q5 + gl It is then clear that h E W (f , xl,... , xk, e) and 7r(h) = g. Assertion 2) has been proved. 3. If A and B are disjoint closed sets in X one of which is compact, then it is easy to construct a realvalued continuous function f on X such that f (A) = {0), f (B) = {1}, and f (X) C [0, 1]. Using this remark and repeating almost literally the reasoning in [66, p. 70], we arrive at the following conclusion: if a subspace Y C X is compact, then there is for each realvalued continuous function g on Y a realvalued continuous function f on X such that fly = g. This means precisely that ir(CC(X)) = C,,(Y). 4. Since X is normal and Y is closed in X, every function g E CC(Y) can be extended to a function f E C,(X). Thus, ir(CC(X)) = C,,(Y). 5. V = X and the continuity of the functions fl, f2 E C,,(X) imply that if f, / f2i then also lily 0 f2ly, i.e. 7r(fl) 7r(f2). By 1), it is continuous.

0.4.2. Remark. Assertion 2) of proposition 4.1 has a converse. The condensation ir: CC(X) -. ar(Cp(X)) figuring in 5) is a homeomorphism if and only if Y = X. It is useful to note the following consequence of proposition 4.1, 5).

0.4.3. Corollary.

Let Y C X, V = X, F C Cp(X), F compact. Then the map

irlp: F - 7r(F) C Cp(Y) is a homeomorphism. The following assertion is obvious (see 0.3.4):

0.4.4. Proposition. If X = Ee{Xa: a E A} is the free sum of topological spaces Xa, then the topological ring Cp(X) is canonically isomorphic to the product jj{Cp(Xa): a E A} of topological rings.

0.4.5. Example.

The space Cp(X, R"0) is homeomorphic to its own countable power, the space Cp(X, R"0)"0, by proposition 3.3. In fact, RR0 is homeomorphic to (R"o)"o.

Also, the canonical homeomorphism between C,,(X, (R"0)"0) and (C,,(X, R"o))"o is an isomorphism of topological rings. On the other hand, proposition 4.4 implies that (Cp(X,R"0))"0 is always (i.e. for any X) homeomorphic as a topological ring to the space Cp(X x N+, R"0). Hence, the fact that the rings CC(X, R"0) and Cp(Y, R"0) are topologically isomorphic does not imply that X and Y are homeomorphic.

The restriction map is dual to the following construction. Let f : X Y be a map (between sets X and Y). Define the map fl: Ry -> RX (between topological spaces) dual to f as follows: if 0 E Ry, then f t (O)(x) _ 4(f (x))

4. RESTRICTION MAP AND DUALITY MAP

13

for all XE X, i.e. fd(¢)=,o f. 0.4.6. Proposition. 1. The map fl is continuous. 2. If f (X) = Y, then fl: RX ---- RY is a homeomorphism from RY onto the closed subspace f 0 (RY) of RX .

Proof. 1. Let f 0 (qi) = z/i and let W(I/', XI, ... , Xk, e) be a standard neighborhood of r/i in RX. For y; = f(xi), i = 1,... ,k, we have fO(W(¢,yl,...,yk,e)) C W (?P, xI,... , xk, e). Hence fl is continuous.

02. Take y E Y such that ¢I (y) # For x E f-'(y) we have f0(0I)(x) = 01(y) # 02(y) = f°(02)(x). Hence

Let f (X) = Y and .01, 02 E R1, 46I

2.

02(y).

f'(0I) # f'(02), i.e. the map fa: RY -> fa(RY) is bijective. Its inverse map (fe)-' is continuous. In fact, if V' = f"(0), then for an arbitrary standard neighborhood U = W(z/), xl, , Xk, e) of r' in RX we have (f)-' (Un f 0(RY)) C W(0, f (xI ), ... , f (xk), f). Clearly, fO(RY) = (0 E RX: if P XI) = A X2), then 41(xI) = ¢(x2)} is a closed set.. The following assertion is of fundamental nature.

0.4.7. Proposition. Let f : X - Y and g: X -> Z be maps onto, i.e. f (X) = Y and g(X) = Z. Then the following conditions are equivalent: 1) fo(C(Y)) C gt(C(Z)); 2) there is a continuous map h: Z Y such that f = hog. 1). Let 0 E fd(C(Y)), i.e. 0 = I/' o f for some I,b E C(Y). Then hl (V)) _ I/ioh E C(Z), since his continuous. But g"(ha(r/i)) = h'(IG)og = Ibohog =T/io f = 0, i.e. 0 E go(C(Z)). We show that 1) 2). Let fd(C(Y)) C g0(C(Z)). We first show that: a) if x E X, A C X, and g(x) E g(A), then f (x) E f (A).

Proof. 2)

Suppose f (x.) # f (A). There exists a function 0 E C(Y) such that 0(f(x)) = 1

and 0(f(A)) = {0}. Then fl(0)(x) = 1 and fa(¢)(A) = {0}. By 1) there is a function ib E C(Z) such that gl(V)) = fa(41). We have i(g(x)) = gl(r1)(x) = 1 and I/,(g(A)) = g*(I/,)(A) = {0}. This contradicts the continuity of I/'.

We now show that g 'g(y) C f'f(y) for any y E X. Let x E g-'g(y). Put A = {y}. Clearly g(x) = g(y) E g(A). By a), f (x) E f (A) = f ({y}) = { f (y)}, i.e.

f (x) = f (y). Thus glg(y) C f-' f (y) Put h(z) = f(g-'(z)) for all z E Z. By what has been proved above, the map h: Z --# Y is well defined. Clearly, hog = fog-' o g = f. It remains to show that h is continuous.

Let B C Z, z E Z, and z E B. Put A = g'(B) and take some x E g -'(z). Then g(x) = z f B = g(A), and a) may be applied to x, A. We deduce: f (x) E f (A), i.e. h(g(x) E h(g(A)). But g(A) = B and g(x) = z. Thus h(z) E h(B), which proves the continuity of h.

0.4.8. Corollary. Let f : X

Y be a map with f (X) = Y. Then: 1) f is continuous if and only if fa(C(Y)) C C(X);

14

0. GENERAL INFORMATION ON Cp(X). INTRODUCTORY MATERIAL

2) if f is a quotient map, then f OC(Y) is a closed subspace of Cp(X ); 3) f is bijective and continuous (i.e. f is a condensation) if and only if fl(C(Y)) is everywhere dense in Cp(X); 4) f is a homeomorphism if and only if fd (C(Y)) = C(X).

Proof. 1. Take for Z the space X itself, and for g the identity map from X onto itself. Then apply 4.7.

2. Let f be a quotient map and 0 E fl(C(Y)), 0 E Cp(X). Take some y E Y. Clearly, every function 0 in fa(C(Y)) is constant on f-I (y). Therefore the function 0 is constant on f-'(y). Hence there is a function g: Y -> R such that i/i = g o f, i.e. z/i = f$(g). Since f is a quotient map and /i is continuous, we deduce that g is continuous. So, ip E fl(C(Y)), i.e. the set fl(C(Y)) is closed in Cp(X). 3. Let f be a condensation, 4' E C(X), and W(46, x1 i ... , xk, e) a standard neighborhood of q in Cp(X ). Put y; = f (xi), i = 1,... , k. Since f is bijective, there is a function 0 E Cp(Y) such that i/i(y;) = 0(x;), i = 1,... k. Clearly, fdti E W(4,x1i... ,.xk,e), i.e. fl(C(Y)) is everywhere dense in Cc(X). Let x1 x2, but f(XI) = f(x2) = y. Then for all 0 E fa(C(Y)) we have ¢(xI) _

0(x2). Take a function i' E C(X) for which O(x1) = 0 and Vi(x2) = 1. Clearly, W (O, x1, X2,1/2) n f" (C(Y)) = 0.

4. Let it be known that f I (C(Y)) = C(X). By 3), f is a condensation. Assume that f is not a homeomorphism. Then there is a closed set P in X such that f (P) is not closed in Y. Take a point y E Y \ f (P) such that y E f (P), and fix a function 0 E C(X) such that ¢(P) = {0} and q5(x) = 1, where x is the pre-image of y under f . If q S E RX and f tl = ql, then )(y) = 1 and t,b(f (P)) = {0}; y E f (P) now implies that ip is not continuous. Hence 0 V f a(C(Y)). This contradicts the requirement. Assertion 4.8 has been proved.

The nonsymmetry in the statement of 2) in 4.8 attracts some attention. It turns out that 2) has no immediate converse. This is related to the fact that a quotient space of a Tikhonov space need not be a Tikhonov space (even when all elements of the partition are closed sets). In the study of function spaces, one naturally borders on reasonings and constructions that are fully applicable within the class of Tikhonov spaces. Hence it is expedient to appropriately modify the notions of quotient space and quotient topology. To this end we define below the R-quotient topology and R-quotient maps.

Let f : X -> Y be a map from a topological space X onto a set Y. Then the strongest of all completely regular topologies on Y relative to which f is continuous is called the R-quotient, or real quotient, topology on the set Y (generated by the map

f). A map f from a space X onto a space Y is called an R-quotient map, or a real quotient map, if the topology on Y coincides with the R-quotient topology generated by f [14].

Let f : X - Y be a map from a space X onto a set Y. Put £ = fO(RY) n C(X) and J = {i E FLY: fa(zG) E C(X)}. Clearly, fg(.F) = 6, and £ is closed in C,,(X) since f n (RX) is closed in RX (see 4.6).

Let T be the R-quotient topology on Y generated by f. Using this notation, the

4. RESTRICTION MAP AND DUALITY MAP

15

following assertion holds.

0.4.9. Proposition.

a) The family 13 = {?/r'(U): z/i E .F, U open in R} is a

subbase for the topology T. b) T is the smallest topology on Y relative to which all functions ili E .F are continuous.

c) If T' is an arbitrary topology on Y such that the map f : X -> (Y, T') is continuous and Y c C(Y, TI), then .F = C(Y, V).). d) For the space (YT) we have C(YT) =.F; moreover, .F is the only completely regular topology on Y for which .F = C(Y, T).

Proof. Clearly, assertions a) and b) are equivalent. We prove c). We have (see 4.7)

f*(.F) c fl(C(Y,T)) c C(X) n fl(RY) = fl (Y). Thus, fO(Y) = fk(C(Y,T')), and since fO is injective, Y = C(Y,T'). The map f : X -+ (Y,T) is continuous, hence c) implies the first part of d). The second part of d) is obvious.

Let now T' denote the topology generated on Y by the subbase S. For each

V = ?k-'(U) E 5 (where

E. F and U is open in R) we have that f -(V) =

f-'Vi-'(U) = (f0V,)-I(U) is an open set in X, since f1Tp E C(X). Thus, the map f is continuous from X onto (Y,T'). Clearly F C C(Y,T'). Applying c) we deduce: .F = C(} , T'). However, the topologies T and T' are completely regular, hence T = T'. This immediately implies a) and b). A map f : X -' Y from a space X onto a space Y is called functionally closed if f"(C(Y)) is a closed subset of C,,(X).

0.4.10. Proposition. A map f from a space X onto a completely regular space Y is an R-quotient map if and only if it is functionally closed.

Proof. Let f be functionally closed. The set C(Y) is everywhere dense in RY, and the map fu is continuous. Thus fO(C(Y)) is everywhere dense in f,W'. But fI(C(Y)) C C(X). Thus fI(C(Y)) C C(X) n f"RY = E and fO(C(Y)) is everywhere dense in E c CD(X). Since f is functionally closed, the set fa(C(Y)) is closed in C'(X). Hence fn(C(Y)) = £, which implies by the injectivity of f, that C(Y) =.F. However, Y is completely regular. Proposition 4.9d) now implies that the topology of Y is the R-quotient topology generated by f. We now assume that f is an R-quotient map. By 4.9d), C(Y) = .F, hence fl(C(Y)) = fl.F is a closed subset of Cp(X). Thus f is functionally closed. In the course of proving proposition 4.10 the following fact was established. Let f : X --> Y be a map from a space X onto a set Y. Then there is exactly one completely regular topology on Y in which f"(C(Y)) is a closed subset of C,,(X) (this topology is the R-quotient topology generated by f).

0.4.11. Example. Not every R-quotient map from a Tikhonov space onto another such space is a quotient map.

0. GENERAL INFORMATION ON Cp(X). INTRODUCTORY MATERIAL.

6

By proposition 4.10 this means that for a map between Tikhonov spaces, functional losedness does not imply that it is a quotient map. The corresponding example is onstructed as follows. Consider some quotient map f : X -. Y from a space X onto a pace Y which is not a Tikhonov space but in which any two points can be separated y a continuous function. It is then obvious that with respect to the R-quotient opology on Y generated by f, the map f is an R-quotient map but not a quotient iap.

5. Canonical evaluation map of a space X in the space CpCC(X) Suppose we are given a set X and a family.F C RX. Then for each x E X there is R by the rule g. ,(f) = f (x) for all f E.F. Putting i/ir(x) = gx :)r x E X gives the canonical evaluation map WF: X - RF.

refined a map g. :.F

1.5.1. Proposition.

For any set X and subspace.F C RX the map g2:F

R is

ontinuous.

This follows from the definitions of qx and the topology on RX.

'.5.2. Proposition. For any space X and subspace.F c CC(X) the map t/iF: X y(.F) is continuous.

'roof. Let X E X and let W (gx, f1, ... , fk, e) be a standard neighborhood of the

inction gx in Cp(.F). Since f1 E.F C Cp(X), we see that tP;1(W(gx, f1i... , fk, e)) _ 1(f; 1((fj(x)-c, f,(x)+e)): j = 1,... , k} is an open neighborhood of x in X. Hence 'F is continuous. Instead of WF we will often write tai. A family .F of maps from a space into a set Y is called separating if for two arbitrary

istinct x1, x2 E X there is an f E F such that f(x1) # A X2). A family F of ontinuous maps from a space X into a space Y is called regular if for any x E X and i C X such that x A there is an f E .F for which f (x) f (A). E.g., the family C(X) of all realvalued continuous functions and the family C°(X) of 11 realvalued bounded continuous functions on a Tikhonov space are regular families. t is useful to know the following

1.5.3. Proposition.

For any set X and family.F C RX the family tb,(X) is a

eparating family of realralued functions on F.

).5.4. Proposition. Let F be a subspace of RX. Then: a) if .F is a separating family of continuous maps on X. then t,= contiruou_sly mops X onto the subspace vy(X) of C-..()'):

3) if F is a

fam.il. of r'1J's on X. M''. t ': X - C.,{F`i . r. X and fhe = h:p::e t = t) of

/r(-T). mophiem 'into'. i.e. a htomemm0rthiss>a

CC

and

5. CANONICAL EVALUATION MAP OF A SPACE X IN THE SPACE CpCp(X)

17

Proof a). Since F is separating and x, # X2, there is a map f E F for which f(xl) 0 f(x2). Then g2,(f) 0 9.2(f), hence 9x, 0 g22. t',(X) is bijective. Hence its inverse is defined, p). By a) the map #F: X t(if' : ti-r(X) X. We show that r 5' is continuous. Let x E X and let U be open in X, X E U. There are e > 0 and f E F such that (f (x) - e, f (x) + e) fl f (X \ U) = 0 (since F is regular). Consider the basis open set V = W (gx, f, e) in Cp(F). Clearly V E) gx and

(V fl t/?F(X)) C U. Hence V,.j' is continuous.

Proposition 5.2 and assertion a) now imply that the map /'f: X - tbr(X) C Cp(F) is a homeomorphism. Proposition 5.4 allows us to identify, for any regular family F C Cc(X), the point tbb,r(x) and the point x, and the subspace ti,r(X) of Cp(X) and X. It is natural to call a family of functions F C C(X) generating if the map 1/;: X -#r(X) C CC(.F) is a homeomorphism. Below we will show that every regular family of hinctions is generating; the converse is false, as is easily proven. 5.4 implies

0.5.5. Corollary.

A space X is homeomorphic to the subspace V,(X) of the space CpCp(X) (of the space CC(C,,(X))).

In relation to the canonical embedding of X in CpCp(X) there naturally arises the notion of R-free linear topological envelope of X. Consider the subspace

LL(X) = {aix1 + + A xn E CpCc(X) : x1i... ,xn E X, A1,... ,An E R, n. E N+}, the set algebraically generated by X in the linear space CpCp(X). Clearly, with respect to the natural operations of addition and multiplication by a scalar in CpCp(X), the space Lp(X) is the smallest linear subspace of the linear topological space CpC,,(X) containing X (more precisely, containing tli(X)). So we have

0.5.6. Proposition. Lp(X) is a locally convex linear topological space over the field R; moreover X can be canonically represented as a subspace of Lp(X).

The dual of a real linear space L is the linear space L' of all continuous functionals (realvalued linear functions) on L, endowed with the topology of pointwise convergence.

0.5.7. Proposition. Lp(X) = (Cp(X))'. Proof. Under the canonical identification of X and Vb(X) C CCp(X), an arbitrary point x E X becomes a linear functional on CC(X); more precisely, x(f) = f (x) for all f E Cp(X). Clearly, the functional x is continuous on CC(X). Therefore the Aix1 are continuous linear functionals on CC(X), which implies that all g E Lp(X) are continuous linear functionals on CC(X). It remains to prove

0. GENERAL INFORMATION ON Cp(X). INTRODUCTORY MATERIAL

t8

0.5.8. Proposition.

If q5 E CC,(X) and q5: C(X) -+ R is a linear function, then there are xl,... , xn E X and A , ,. . . , an E R (for some n E N+) such that O_A1x1+-..+Anxn.

Proof. Take f - 0 E Cp(X ). Then ¢(f) = 0 (since 0 is linear), and since 0 is 2ontinuous there are X 1 ,

, xn E X and e > 0 such that ¢(W (f , x1, ... , xn, e)) C

(-1,1). We may assume that xi J xi if i yt j. Let g E C,(X) with g(xi) = 0, i = 1,... , n. We show that then q5(g) = 0. In fact, for each k E N+ we have kg E W (f, x1, ... , xn, e), hence 14(kg) I < 1. By the linearity of 0 we obtain kIq5(g) < 1.

Thus, I¢(g)I < 1/k for all k E N+, i.e. 0(g) = 0. Take gi E CC(X) such that gi(xi) = 1 and gi(x3) = 0 for i # j, i = 1,... n, and put A, = 0(gi). We verify that for an arbitrary g E CC(X), 0(g) = A,g(xi) + of propositions 5.7 and 5.8, since

+Ang(xn). This will finish the proof

A19(XI) A- ... + An9(xn) = (A1xi + ... +.nxn)(9)

Put g' = g - g(x1)g1 - g(xn)gn. Clearly, g' E Cp(X), and g'(xi) = 0 for all i = 1,... , n. Thus, by what has been proved above, 4'(g') = 0. This, taking into account the linearity of ¢, implies 0 = q5(g') = ¢(g) - ,5(E,"_1 g(xi)gi) and

0(9)=O

= Eg(xi)0(9i) = E Aig(xi) (t019) :-1 i=t i=1

0.5.9. Proposition. a) X is closed in LL(X); ,Q) L,(X) is closed in C9CC(X); -y) X is closed in CCCC(X).

Proof. a) Let X and LAX) denote the respective closures of the sets X and L,(X)

in CC,,(X). Let y E X \ X. Put Y = X U {y} C CCC(X), and assume that + Anxn for certain x1,... , xn E X. Then y # xi, i = 1,... , n. Hence there is a function 0 E C,(Y) such that ¢(y) = 1 and O(xi) = 0, i = 1, ... , n. For the function f = 41 x we have xi(f) = f (xi) = O(xi) = 0, and hence y(f) = (Aixi + + + Anxn(f) = 0. Put P = {x E X: f (x) > 1/2}. Since 0 is Anxn)(f) = A1x1(f) + continuous and 0(y) = 1, we have y 0 X \ P and y E P. But x(f) = f (x) > 1/2 for all x E P. The fact that y E P now implies that y(f) > 1/2, contradicting y(f) = 0. 13) By 5.7 the set L,,(X) consists of all functions 45 E CC,(X) that are linear. But the latter set is closed in CCC(X), since the closure of an arbitrary set of linear functions in CC,(X) consists of linear functions only. a) and /3) imply 'y). y = AIXI +

0.5.9'. Proposition. The space X (= t(b(X)) is C-embedded in Rc(r) Proof. Each continuous function f on 1/'(X) = X can be interpreted as the restriction to ip(X) of the projection from the space RC(r) onto the f th factor R1.

5. CANONICAL EVALUATION MAP OF A SPACE X IN THE SPACE CpCp(X)

19

0.5.10. Proposition. Cp(X) = (Lp(X))'; more precisely, the linear topological space Cp(X) is canonically isomorphic to the space (Lp(X))'. The following fact lies at the basis of proposition 5.10.

0.5.11. Proposition. Each realvalued continuous function f on X can be uniquely extended to a continuous linear function f : Lp(X) -+ R.

Proof. We have ?,b(f) E Cp(CpCp(X)), where tJ'(f)(g) = g(f) for all g E CpCp(X); moreover, ? (f) is a continuous linear functional on CpCp(X) D Lp(X). By restricting tk(f) we find the required extension f (clearly, f iX = f)- The uniqueness of f follows from the linearity of f and the fact that every y E Lp(X) can be written as

y = AIxl +

+ Anx" with x1t... x" E X. Thus, as regards proposition 5.10 it

has been proved that there is a canonical bijective map from Cp(X) onto (Lp(X))'. Without difficulty it can be shown that this map is linear and continuous together with its inverse. We can also reason differently: rli(CC(X)) C CpCPCp(X), and if g E (Lp(X))', then

g1x E Cp(X) and g E t/1(Cp(X)). Thus (Lp(X))' = t//(Cp(X)). But Cp(X) can be canonically identified with tli(Cp(X)), allowing us to write (Lp(X))' = Cp(X).

0.5.12. Corollary. Lp(X) is linearly homeomorphic to Lp(Y) if and only if Cp(X) is linearly homeomorphic to CC(Y).

0.5.13. Proposition. Let P be any class of topological spaces having the following properties:

1) the image of a space in P under a continuous map belongs to P; 2) if X = U{X;: i E N+} and Xi E P for all i E N+, then also X E P;

3) ifX EPandYEP, then ahoXXYEP; 4) REP. If X E P, then also Lp(X) E P.

Proof. Let X EP. Properties 3) and 4) readily imply that also X" X R" E P for all

nEN+.

We define the map ¢n: X" X R" Lp(X) by: 0n(xli... x,,al, )an) = a1x1 + + anxn, and put Ln = q5n(X" x R") C Lp(X). The map 0n is clearly continuous,

hence L" E P (by 1)). It remains to note that Lp(X) = U{Ln: n E N+}, and to apply 2).

0.5.14. Corollary. Let a space X have one of the following properties: 1) X is a-compact (i.e. X is the union of a countable family of compacta); 2) X is a Lindclof E-space in the sense of K. Nagami [123]; 3) Xn is a Lindelof space for every n E N+; 4) X is separable.

0. GENERAL INFORMATION ON Cp(X). INTRODUCTORY MATERIAL

20

Then the space Lp(X) has the same property.

The following notation will be often used in the sequel. Let y E Lp(X). If y = 0 is the zero element of the space Lp(X), then we put l(y) = 0. If y ¢ 0, then 1(y) = min{n E N+: there are xl,... , xn E X and A1, ... , An E R. such that y = AIx1 +

+ Anxn}. Further,

Lp.,(X) = {y E Lp(X) : l(y) < n}

0.5.15. Remark.

and

Mpr.(X) = Lp.,(X) \ Lp--,(X).

Proposition 5.11 and the fact that X is a Tikhonov space

readily imply that X is an algebraic basis of the linear space Lp(X ), i.e. each nonzero

y E Lp(X) can be uniquely (up to order of terms) written as y = Aix, +

+ Anxn,

where A,,...,A ER\{0},xl,...,xnEX, and n E N+.

0.5.16. Proposition. The set L,, (X) is closed in Lp(X) for every n E N+. Proof. The set Lpo(X) = {B} is closed. Let n > 1 and y E L9(X) \ Lpn (X). Then

k>n,Ai#0,i=1,...,n,and x,#x;for i#j. Take

pairwise disjoint neighborhoods V of the points x1, i = 1,... , k, and fix functions fi E Cp(X) such that fi(xi) = 1 and fi(X \ [,) = {0}. By 5.11, the functions fi can be extended to continuous linear functions fi on Lp(X). The set u = fl{ f i ' (R\ {0}): i = 1,... , k} is open in Lp(X), and y E U since fi(y) = Ai # 0 for each i = 1,... , k. We show that U fl 0, i.e. that l(z) > n for all points z E U. Let 1(z) = m, z = µ1x', + ... + µ,nx',,,, where µi # 0 and x; x'; if i j. The membership z E U implies that i(z)34 0. Thus, Therefore U fl {x ... , x;n} # 0 (for every i = 1,... , k). Put Vi' = U fl {x ... , x;n}. The system {U': i = 1,... , k} consists of pairwise disjoint nonempty sets. Thus m > k > n, i.e. l(z) > n and U fl Lp.,(X) = 0.

0.5.17. Proposition. Let y = A1x1 +

+ Anxn E Mpn (X) = Lpn (X) \ Lp.,-t (X) -

Then the family o f all sets o f the form O(VI, ... , V n , e) = {y' = ix', + + A x,: IAk - AkI < C, xk E Vk, Vk open in X, k = 1, ... , n} is a base at the pointy in the space Lp.. (X) .

Proof. Put Ai = {A E R: IA' - Ail < e}. Clearly, O(V1i... ,Vn,e) = AIV1 +

+

AnVn. The operations of addition and multiplication are continuous in Lp(X ), hence it suffices to prove that every O(V1 i ... , V,,, e) is an open set in Lp. (X ). We may restrict

ourselves to the case when 0 V Ai for all i = 1,... , n and the family V1,... , Vn is disjoint. Let z

= uIxi + + Ax' An X' E O(V1 >... , Vn, e). Fix fi E CP(X) i = 1 ,... , n, 0). Then z(fi) _ such that fi(x'i) = 1/µi and fi(X \ U) = {0} (clearly, the µi µ1xc (fi) + + µnx'n(f) = µix'i(fi) =N(1/110 = 1. Since pi E A,, there exist open sets 0' and 02 in R such that 1 E 0', 1/µi E 0'i 0 0 0', and 0'(02')-' C Ai. Take functions gi E Cp(X) for which gi(x'i) = 1 and gIX\f, i(Oi) = 0, i = 1,... n.

5. CANONICAL EVALUATION ASAP OF A SPACE X IN THE SPACE CpCP(X)

21

The set I4' = { f,'(O;) fl g -'(R \ {0}): i = 1,... n} is open in L,(X). The fact that gi(x) 34 0 implies that fi(x) E 02, fi(x) # 0, x E V. Hence gi(x'j) = 0 for j i, and by using the disjointness of the family V1, ... , V we obtain that + µ 9;(xn) = pigi(x'j) = µi # 0. Moreover, fi(z) = 1 on the §i (Z) _ lA19i(xi) + same grounds. Hence z E W. As in the proof of assertion 5.16, we can easily show that if z E W, then l(i) > n. Hence W fl L. (X) C Mr. (X). Consider an arbitrary z = 1ii + + a. in E 14' fl Lpn(X). We have f,(i) 36 0 for all i = 1,... , n. We may assume that the system al, ... , in is enumerated in such a way that fi(i,) # 0 and ii E V,,. We show that also )1; E A1, i = 1,... , n.

We have gi(i) = aigi(xi) and gi(i) # 0. Hence ii E fi '(02), i.e. fi(ii) E O. By the definition of W, fi(i) E 0;. Moreover, fi(i) = ai f1(i,) (see above). So .Ai fi(ii) E O. This implies 5ifi(ii) (fi(i,))-' E O'1(O2)-' C A1, i.e Ai E Ai for all i = 1,... , n. Hence z E W fl LP,. (X) C A1V1 + + V implying that the set in L,. (X). Let. 1, be an arbitrary (locally convex) linear topological space over the field R. In L there is a Hamel basis: a subset B C L such that every finite subset of B is linearly independent and every vector in L can be written as a linear combination of finitely many vectors in B. Clearly, the space L" algebraically dual to L, consisting of the realvalued linear functionals on L, is in canonical bijective correspondence with the set RB of all realvalued functions on B. This correspondence (realized by restricting a ¢ E La to B) is clearly a homeomorphism if LI is endowed with the topology of pointwise convergence on L (the space thus obtained will be denoted by LP) and RB is given the topology of pointwise convergence on B (i.e. the product topology). On the other hand, the space LP of all continuous linear functionals on L (in the topology of pointwise convergence on L) is everywhere dense in L.O. Thus the space LP has all the topological properties present, in the topological space RB that are inherited under transition to an everywhere dense subspace. In particular, we have,

0.5.18. Theorem [65]. Let L be a linear topological space over R and Lp the space of all continuous linear functionals on L in the topology of pointwise convergence. Then every uncountable regular cardinal is a precaliber of L'p, and L'p is perfectly-r.normal.

Assertions 5.18 and 5.7 imply:

0.5.19. Corollary. For each space X the space LL(X) has the following properties: 1) every uncountable regular cardinal (in particular, 111) is a precaliber of Lp(X); 2) the Suslin number of LD(X) is countable; 3) the space Lp(X) is perfectly-n-normal.

0.5.20. Proposition. If a space X is Dieudonne complete but not Hewitt-Nachbin complete (such is, e.g., a discrete space of Ulam measurable cardinality), then the space Lp(X) is not Dieudonne complete.

0. GENERAL INFORMATION ON Cp(X). INTRODUCTORY MATERIAL

22

Proof. Assume the contrary. Then c(Lp(X)) < Ro implies that Lp(X) is HewittNachbin complete. We conclude that X is also Hewitt-Nachbin complete.

6. Nagata's theorem and Okunev's theorem Already in §1 we drew attention to the fundamental significance of the following fact.

0.6.1. Theorem (J. Nagata [1241). If the topological rings CC(X) and CC(Y) are topologically isomorphic, then the spaces X and Y are homeomorphic.

Proof. A functional g: CC(X) -- R is called multiplicative if g(f h) = g(f) g(h) for all f, h E Cp(X). We denote by X the subspace of CCc(X) formed by all nonzero continuous linear multiplicative functionals on Cp(X). Clearly, X C X C Lp(X). The topological isomorphism between Cp(X) and CG(Y) can be extended in an ohvions manner to a homeomorphism between k and Y. Therefore Nagata's theorem will be proved if we establish that X = X and Y = Y (more precisely, k = ii(X) and Y = V)(Y)). It suffices to establish that k C X.

Let g E X. By requirement g 0 0. Now g E Lp(X) \ {©} implies that there are

andxi0x;ifi#j.

andA,i...,A R\{0} for which

Case 1. Let n > 1. Take fl,f2 E CC(X) such that f1(xi) = 1/A1, f2(x2) _ 1/A2, and fi(x,) = 0 for the remaining values of i = 1, 2, j = 1,... , n. Then AIf1(x1)+...+Anf1(xn) = 1 and g(f2) = 1, but g(fl) =

g(fl f2) = 0 since fl f2 - 0. Hence case 1 is impossible.

Case 2. Let n = 1, i.e. g = Ax1. Since g 0 0, Al # 0. Take the function fo E Cp(X )

identically equal to 1. Then fo = fo and 9(fo) = g(f02) = g(fo) g(fo) On the other hand, g(fo) = A1x1(fo) = AIfo(xi) = Al. We obtain ai = A. It follows from A t 0 that A = 1, and g1 = .TI E X. The theorem has been proved. In another direction there is the following theorem of O. G. Okunev, which, like Nagata's theorem, distinguishes itself by its great generality.

0.6.2. Theorem.

Let X = Y x R, i.e. the space X is the product of a space Y

and the real line R. Then the space (Cp(X))'o is linearly homeomorphic to the space CC(X).

Proof. Let Z be the discrete space of integers, Z C R, and X0 = Y x Z C X. For an r E denote by jr] its entier, i.e. the largest integer not exceeding r.

Further, let (r) = r - [r] be the fractional part of r. For x = (y, r) E X we put x = (y, [r]) E Xo and x+ = (y, [r] + 1) E Xo. Cp(X ), i.e. the Let as construct the continuous extension operator 1S: Cp(Xo) continuous linear map 0: Cp(Xo) -+ Cp(X) such that for each function f E C(Xo) the restriction of the function 0(f) E C(X) to X0 coincides with f. So, let f E CC(Xo)

6. NAGA'rA'S THEOREM AND OKUNEV'S THEOREM

23

and x = (y,r) E X. Put b(f) (y, r) = 0(f) (x) = (r) ' f (x+) + (1 - (r) )f (X-)

If r E Z, then (r) = 0, x- = x, and 0(f) (x) = f (x.), i.e. the restriction of ¢(f) to X0 coincides with f. It, is easy to verify the continuity of O(f). Finally, the linearity of the map ¢: C(Xo) ---* C(X) is obvious. Since the value of ¢(f) at an arbitrary point x E X depends only on the values of f at the two points x and x+, the map q5 is continuous with respect to the topology of pointwise convergence on C(X) and on C(Xo). Thus, .0: CP(Xo) -+ CP(X) is a continuous extension operator. Therefore the space CP(X) is linearly homeomorphic to the space CP(Xo) x L, where L = {g E CP(X): g(,Yo) = {0}}. In fact, the map V': CP(Xo) x L -+ CP(X)

defined by the rule tp((f,g)) = 4(f) + g E CP(X) for all (f,g) E CP(X) x L, is clearly a continuous linear map. It is easily verified that ip is also bijective, and that.

b'(CP(X) x L) = CP(=C) (if h E CP(X), f = hl .-<,, and 9 = h - 0(f), then g c- L, f E CP(X0), and V'((f,g)) = h). Since X0 is the free topological sum of NO copies of Y, the spaces (. p(X) and (CP(Y))"0 are linearly homeomorphic (see chapt. 0).

For nEZwe put Ln = {g E CP(Y x [n,n + 1]) : g(Y x {n}) = g(Y x {n + 1}) = {0}}. Taking for each n E Z an arbitrary function in Ln, we obtain a function in L. Hence it is clear that L is linearly homeomorphic to the product fj{Ln: n E Z}. However, clearly all Ln are linearly homeomorphic to Lo. Hence L is linearly homeomorphic to (Lo)"'. If we denote the relation /of linear homeomorphism byth , we can now write: C'P(X)

N

( (,Yo) x L -" (CP(Y))"0 x LnO

-h

(C,(Y) X L0)K° t^ ((CP(Y) X f'o)"O)"0 1h (CP(X))N°

Theorem 6.2 has been proved.

0.6.3. Corollary.

If X is homeomorphic to some locally convex linear topological space over the field R and IXJ > 1, then the spaces CP(X) and (CP(X ))E0 are linearly homeomorphic.

Proof. It is well known that every nontrivial locally convex linear topological space X is homeomorphic to a space of the form Y x R, for some space Y. Corollary 6.3 is, in particular, applicable to spaces of the form CP(X). We obtain

0.6.4. Corollary.

For any nonempty space X, the space CPCP(X) is linearly

homeomorphic to its countable power.

CHAPTERI

Topological properties of CC(X) and simplest duality theorems

1. Elementary duality theorems In this section we have gathered the simplest general theorems in which properties of a space X are characterized by a topological property of Cp(X ).

I.1.1. Theorem. For any space X, 1 X I = X(C,(X)) = w(CC(X))

Proof. We have X(Cc(X)) < w(Cc(X)) < w(RX) < X. Let us prove that IXI < X(Cp(X)). Assume the contrary and fix a base ry of the

space Cp(X) at the point f = 0 such that try) < XJ. We may assume that. all elements of y are standard open sets in C,,(X ). For each W(f, x1.... , Xk, e) E ry

we let K(W) = {x1,... and put Y = U{K(W): W E y}. Then I}'I < JXI, and there is a point x` E X \ Y. Put U = W (f, x`,1), and consider an arbitrary V = W (f , x1.... , xk, f) E y. We have x1 i ... , x E Y, hence xi # x' for i= 1,... , k. There is a function g E CC(X) for which g(xi) = 0, i = 1,... ,k, and g(x*) = 1. We obtain that g E V \ U, i.e. V \ U L 0 for all V E y. This contradicts the fact that y is a base of CC(X) at the point f E U. Theorem 1.1 implies that a space X is countable if and only if CC(X) satisfies the second axiom of countability.

1.1.2. Corollary. If a space X is uncountable, then the space CC(X) does not have a countable base.

In particular, there are no countable bases in the space CC(R) and Cp(I). On the other hand, there are countable spaces without countable bases [661. For such a space

X the weight of Cp(X) is countable, and hence less than the weight of Y. Thus, the weight of X can be smaller or larger than the weight of Cp(X). Therefore the following duality theorem for the network weight is of special interest [11, [141. 25

26

1. TOPOLOGICAL PROPERTIES OP Cp(X) AND SIMPLEST DUALITY THEOREMS

1.1.3. Theorem. For any space X, nw(X) = nw(Cp(X)).

Proof. We show that nw(CC(X)) < nw(X). Fix a network P in X and a countable base B in R. For each pair of collections S1i ... , Sk E P and U1,... , Uk E B we

fix W(S1i... ,Sk,U1,... ,Uk) = {f E C(X): f(Si) C Ui, i = 1,... ,k}. Let y = ,Uk): Si)... Sk EP, Ul,...,Uk E [3). We show that -y is a network in CC(X). This will imply the required inequality, since 171

IPI Let f E CC(X) and let W (f , xl, ... , Xk, e) be a standard neighborhood of f in CC(X). Assume that xi # x; if i # j. Choose sets U1,... ,Uk E B open in R such that f (xi) E U; C (f (xi) - e, f (xi) + e) for i = 1, ... , k. Since f is continuous, there

are S1,... , Sk E P such that xi E Si and f (xi) E U1 for i = 1, ... , k. We show that f E W (S1i ... , Sk, U1, ... , Uk) C W(f, xl,... , xk, e). The membership follows from the fact that f (Si) C U1, i = 1, ... , k. Let g E W (S1, ... , Sk, U1, ... , Uk). Since xi E Si, g(xi) E Ui; hence Jg(xi) - f (xi)J < e for all i = I,_ , k.. Thus g E W(f,x1,... ,xk,e). The opposite inequality can be proved in a rather simple manner. The inclusion X C CCC(X) and the inequality already proved imply that nw(X) < nw(CpCp(X)) < nw(Cp(X)). Hence nw(X) = nw(Cp(X)).

1.1.4. Theorem. Always d(X) = iw(Cp(X)) = ,i(Cp(X)) (see [14J).

Proof. It suffices to establish that iw(Cp(X)) < d(X) ?P(CC(X)), since iw(Cp(X)) > t/'(Cp(X)). Put r = d(X) and take Y such that Y = X and JYI < r. Then w(Cp(Y)) _< w(R) < r. The restriction map Try: Cp(X) Z C C,(Y) is a condensation from Cp(X) onto the subspace Z = iry(Cp(Y)) of the space Cp(Y) (chapt. 0, proposition 4.1). We have w(Z) < w(Cp(Y)) < r, and hence

iw(Cp(X)) < w(Z) < r = d(X).

We show that d(X) < iG(CC(X)). Take f E CC(X), f =_ 0, and fix a family -y of standard neighborhoods of f in Cp(X) such that fl-y = {f}. For each W (f , x1i ... , xk, e) E y we put K(W) = {x1,... , Xk}, and we consider the subspace Y = U{K(W): W E -y} of X. Clearly, JYJ < 17J. We show that Y = X. Assume the contrary. Then there are a point x` E X \Y and a g E Cp(X) such that g(x) = 1 and gly =_ 0. Then g E fly and g # f, a contradiction.

1.1.5. Theorem. iw(X) = d(Cp(X)) [126].

Proof. It follows from X C CpCp(X) that iw(X) _< iw(CpCp(X)). By theorem 1.4, iw(CpCp(X)) = d(Cp(X)). We show that d(Cp(X)) < iw(X). Let f : X Y be a condensation and w(Y) < iw(X). By theorem 1.3, nw(Cp(X)) < w(Y); hence

27

1. ELEMENTARY DUALITY THEOREMS

nw(ftl(CC(}')) < w(Y), since f d is a homeomorphism into (chapt. 0, §4). Assertion 4.8 in chapt. 0 implies that fa(C,,(Y)) = CC(X), therefore

d(Cp(X)) < d(fd(Cp(Y))) 5 nw(ftl(Cp(Y))) < w(Y) < iw(X). Theorems 1.1, 1.3, 1.4, and 1.5 imply

1.1.6. Corollary. Let X t Y, i.e. Cp(X) is homeomorphic to Cp(Y). Then: a) nw(X) = nw(Y); b) d(X) = d(Y); c) iw(X) = iw(Y); d) IXI = IYI. In other words, the network weight, the density, the i-weight, and the cardinality are supertopological invariants.

We now give an example of l-equivalent spaces from which it can be seen that. it large number of important topological properties is not only not preserved tinder t-equivalence, but also under l-equivalence.

1.1.7. Example.

Let X = {(n,m): n,m E N}, X = {(n,m): in E N), n E N,

where the topology on X is determined by the requirements: 1) every Xn is open in X; and 2) each Xn is a compactum whose only nonisolated point is (n, 0). Thus, X is the free sum of countably many copies of an ordinary convergent sequence. Consider the partition of X whose only nonsingleton element is the set F = {(n,0): n E N) of all nonisolated points in X (F is closed in X). The quotient space corresponding to this partition will be denoted by Y. Thus, Y is the so-called `nonmetrizable countable hedgehog'. Y contains the unique nonisolated point F. The other points of Y will be denoted by the same symbols as the corresponding points in X, i.e. by (n, m). An arbitrary function f E C9(X) can be put into correspondence with the map Of: N x N -+ R defined by: Of (n, m) -

f (n, m) - f (n, 0) f (n, 0)

if m 340, n E N; if m = 0, n E N.

Let Zo C RN"N be the image of C9(X) under the map f -+ of. An arbitrary function g E Cp(Y) is put into correspondence with the map 719: N x N -- R defined by: X19(0, 0) = g(F), ?19(n, 0) = g(n - 1,1) for n E N+, and i/i9(n, m) = g(n, m +

1) - g(F) if m E N+, n E N. Put O(f) = of for f E Cp(X ), and 7j'(g) =

9

for

g E CC(Y). Clearly, 0 linearly and homeomorphically maps CC(X) onto the linear topological space Zo of all infinite matrices {anm: n, m E N} of real numbers for which each row {anm: m E N} is a sequence converging to zero, where Zo has the topology

of elementwise convergence, i.e. is regarded as a subspace of the product R!"'. It can be readily verified that ib linearly and homeomorphically maps C,,(Y) onto Zo. Hence CC(X) and CC(Y) are linearly homeomorphic.

28

I. TOPOLOGICAL PROPERTIES OF Cp(X) AND SIMPLEST DUALITY THEOREMS

So we have countable spaces X and Y with linearly homeomorphic spaces CC(X) and CC(Y), and having the following combination of properties: X is locally compact, has a countable base, is metrizable, and has infinitely many nonisolated points;

Y is not locally compact, does not satisfy the first axiom of countability, is not metrizable, has uncountable weight, and has only one nonisolated point.

1.1.8. Example.

Let Z = X U {C}, where C 0 X and the topology on Z is such that the above described space X is an open subspace of Z. If Z D V D C, then V is open in Z if F \ V is finite and V fl X is open in X (where F is, as before, the set {(n,0): n E N} of all nonisolated points of X). This space is well known [16], [3]. It is a sequential space but not a Frechet-Urysohn space. For f E CC(Z) we define gf : N x N+ - R by: g f(n, m) = f (n, m) - f (n, 0) if (n, m) E N+ x N+, and g f(O, m) = f (m - 1,0)-f(C) if m E N+. Consider the linear topological space L3 in RN"N+ consisting of all g: N x N+ -+ R for which g(n,m) --+ 0 as in L3 x R oo for all n E N. The neap 0: C,,(Z) defined by 0(f) = (g f, f (t:)) is, as is readily seen, a linear homeomorphism. Thus, the spaces CC(Z) and M3 = L3 x R are linearly homeomorphic. We now note that M3 is linearly homeomorphic to L3 (see the definition of t/ig), while the latter is linearly homeomorphic to Zo. We arrive at the following conclusion. 1.1.9. Corollary. The weight, the character, local compactness, the Pr chetUrysohn property, and metrizability of a space are not preserved under linear homeomorphisms (let alone under homeomorphisms) of the function space over it, i.e. are not only not supertopological, but even not 1-topological properties.

2. When is the space Cp(X) v-compact? If X # 0, then C,,(X) is not compact. Only very seldom does the space Cp(X) have the property of a-compactness: only in trivial cases.

1.2.1. Theorem (N. V. Velichko). The space Cp(X) is a-compact if and only if the set X is finite.

V. V. Tkachuk and D. B. Shakhmatov have proved that X is finite also in case Cp(X) is a-countably compact [58]. We now state a general result from which these two assertions follow [12].

1.2.2. Theorem.

Let Y be everywhere dense in X. If CP(YIX) is o-countably compact, then X is pseudocompact and Y is a P-space (see chapt. 0, §4). Recall that a P-space is a space in which all sets of type Ca are open.

Proof of theorem 1.2.2. Let Cp(YIX) = U{Zi: i E N+}, with all Zi countably compact.

2. WHEN IS THE SPACE Cp(X) a,-COMPACT?

29

We show that X is pseudocompact. It suffices to establish that there does not exist an infinite discrete family of nonempty open sets in X. Assume the contrary: let there be such a family l; = {Ui: i E N+}. Then UinY # 0 for all i E N+, since Y is everywhere dense in X. Fix yi E U; n Y for i E N+. The set B; = { f (yi): f E Z;} is bounded in R, since B; is a continuous image of the countably compact space Zi. Hence B; # R. Fix a; E R \ B; for i E N+. Since is a discrete family in X, there is a continuous function g on X such that g(y;) = ai for i E N+. Then g(yi) V B;, hence gly Z, for all i E N+, i.e. gay U{Zi: i E N+} = Cp(YIX); a contradiction. So X is pseudocompact. We now show that Y is a P-space. Assume the contrary. Then there are a point y' E Y and a countable family {1 i E N+} of dosed sets in Y such that y' F; C Fi+I for i E N+, and y' E U{F; : i E N+}. The subspace Z' = {f E Cp(YIX): f(y*) = 0} is closed in CC(YIX). Hence Z' is o-countably compact: Z' = U{ZZ.: k E N+), where every Zk is countably compact. With these notations we have:

1.2.3. Lemma.

Fix an arbitrary e > 0 and a k E N+. Then there is an ik E N4 such that for each function f E Zk there is a point yf E Fik for which f (yf) < e.

Proof. Assume the contrary. Then for each i E N+ we can choose a function f E ZZ such that f (y) > e for all y E F. Since Zk is countably compact, there is a function f E Zk C CC(Y) which is a limit of the sequence { fi: i E N+} in the topology of CC(Y). It is obvious that f (y) > e for all y E U{ F;: i E N+} (we use, in particular, that Fi C F;+I). Since f is continuous and y' E U{F; : i E N+}, we have f (y') > c > 0. But f E Zk C Z', hence f (y') = 0. This contradiction completes the proof of the lemma.

We continue with the proof of the theorem. For each k E N+ and e = 2-k we choose a number ik E N+ in accordance with the lemma.

There exists on X a realvalued continuous function gk such that gk(y') = 0, Igk(x)t < 2-k for all x E X, and gk(x) = 2-k for all x r= Fjk. We then define

the continuous function g: X -+ R by g = E{gk: k E N+}, i.e. g(x) = E{gk(x): k E N+} for all x c- X. In particular, g(y') = E{,qk(y*): k E N+} = 0, and hence gly E Z. However, for *each k E N+, by definition g(y) > 2-k for all y E F. Thus, for all k E N+, gay 0 Zk by the choice of ik. We obtain that gly 0 U{Zk: k E N+}; a contradiction. The theorem has been proved. Theorem 2.2 has no converse. To get convinced, take Y to be an uncountable discrete space and X the one-point compactification of the space Y. Then the space Cp(YIX) contains as a closed subspace the E.-product of JYJ copies of the real line (chapt. 4, §5). As can be readily verified, every pseudocompact P-space is finite. Hence theorem 2.2 implies

1.2.4. Corollary [58]. If the space CC(X) is a-countably compact (in particular, if it is o-compact), then the space X is finite.

30

I. TOPOLOGICAL PROPERTIES OF Cp(X) AND SIMPLEST DUALITY THEOREMS

The conclusion of 1.2.4 is no longer valid if CC(X) is assumed to be merely or-pseudocompact.

1.2.5. Example [171, [1451. D. B. Shakhmatov has constructed for an arbitrary cardinal r > 2110 an everywhere dense pseudocompact space XT in F such that for any countable set A C X, and function f : A -> I (no continuity assumption), f can be extended to a realvalued continuous function on all of X,. The unit ball in Cp(X,) is pseudocompact, since under projection of IX- onto a Vi = countable facet of it, the set VI is mapped onto this facet (see the lemma below). Hence Cp(X,) is o-pseudocompact. We now show how to construct X,. Let M denote a minimal well ordered set of cardinality 2140, and let IM = H{Ia: a E M} be the Tikhonov cube of weight 2N0. For B C M we denote the natural projection IM -+ IB by Ir83A

key role is played by the following lemma.

I.2.6. Lemma.

Let X be a dense subset of IT. Then X is pseudocompact if and only if lrB(X) = IB for all countable B C T.

Proof. In fact, if 7rB(X) = IB for all countable B C r, then for f c- CC(X) we have f = 9 0 IrB0, where (Bob = No and g E CP(IBo). Since IBo is compact, f (X) = g(IBo) is bounded in R; thus X is pseudocompact. Conversely, if X is pseudocompact, then lrB(X) is everywhere dense in IB, and iB(X) is compact since w(lrB(X)) = 11o. Thus 7rB(X) = IB for all countable B C r, and the lemma is proved. Let G = {x E IM: I {a E M:7ra(x) # 0} 1 < tto} C IM be the E-product with center

at zero. Then IGI = 2"0 = (MI. Let {ga: a E M} be an enumeration of the elements

.ofGsuch that I{aEM: g=ga}l =2HoforallgEG. LetI= {Ac M: JAI :5 No). Clearly, IEI = 2n0. We choose an enumeration {Ap: ,8 E M} of the elements of E similar to that for the elements of G: each set A E E occurs in it 2N0 times. For each

aEMwe determine apoint X.E1Mby: 1

if 7 < a; if -y > a, a E A,;

to

if y>a,or ¢A,r.

N7(9a)

7r7(xa) =

The space X, = X = {xa: a E M} C IM is the one we looked for. We show that for any countable B C M we have lrB(X) = IB. By the lemma, this will prove the pseudocompactness of X. Let g E IB be arbitrary. There is an a > sup{b: b E B} such that g = 7rB(ga). By construction, NB(xa) = g, as required. We show that all countable subsets of X are closed and C°-embedded. At first we will convince ourselves that for any countable set B C M the space clJM({xa: a E B}) is homeomorphic to #N. It suffices to prove that for all countable subsets MI, M2 C M

such that MI f1M2=0we have clju({xa: aEMl})lcl1M({xa: aEM2})=0. Let 0 E M be such that 9 > sup(MI U M2) and Ae = MI. Then 7re(xa) = 1 if

aEMI,and7re(xa)=0ifaEM2. Thus the sets Ix.: a E M1} and {x: aEM2}

3. L'ECH COMPLETENESS AND THE BAIRE PROPERTY IN SPACES Cp(X)

31

are functionally separated in I". We have also shown that all countable subsets of X are closed in X.

Finally, let B C M be countable and f E Cp({xa: a E B}, I) = IB.

Since

P = cl!M({xa: a E B}) is homeomorphic to QN, there is an fo E Cp(P,I) such that fo(txa:aEB) = f. Clearly, there is an fl E Cp(IM,I) such that f1(p = fo. Then f = f I (X is the required function on X. In 11511, V. V. Tkachuk obtained a number of results going in the direction of 1.2.3.

In particular, he clarified when C.'p(X) is o-pseudocompact and when CC(X) is abounded, and considered similar questions for the space Cp (X) of bounded continuous functions on X (see also 112), 1581).

3. Cech completeness and the Baire property in spaces CC(X) Compactness can be regarded as the highest absolute form of completeness of a space-compacta are closed in any ambient space. With respect to the topology of uniform convergence, the space. C(X) is always complete; this is one of the basic principles of functional analysis. The following question naturally arises: when is the space CJ(X) complete? However, since the topology of C,(X) is, as a rule, not metrizable, this question needs to be made more precise: what is to understood under completeness?

If completeness of CC(X) is understood as completeness relative to the natural uniform structure (induced by the topology of CC(X) and the group structure of (CC(X)), the answer is simple and noninteresting: Cp(X) is complete if and only if X is discrete. An important version of topological completeness is Cech completeness. A space is called Cech complete if it is a set of type Ga in some (hence in any) Hausdorff compactification of it. For metrizable spaces Cech completeness is equivalent to metrizability

by a complete metric-this alone makes clear the importance and usefulness of the notion of Cech completeness. Further, every Cech complete space that is similar to a complete metric space has the Baire property, i.e. in it every countable family of nonempty everywhere dense open sets has nonempty intersection. Below we will show that with respect to Cech completeness and the Baire property the space Cp(X) fundamentally differs from the space C(X) endowed with the topology of uniform convergence. We will prove that only in trivial cases the space Cp(X) can be Cech complete, and that it almost never has the Baire property.

1.3.1. Theorem. If the space CC(X) contains an everywhere dense Cech complete subspace, then X is discrete and countable.

Proof. Let Z C Cp(X) C RX, and let Z be tech complete and everywhere dense in Cp(X). Then Z is everywhere dense in RX as well, since Cp(X) = Rx. Hence Z is a set of type G6 in RX [16]. We show that CC(X) = RX.

Let9ERX. The mapip:RX ->RXisdefinedbyOf )= f+g for all f ERX. Clearly,

' is a homeomorphism of Rx onto itself.

Hence the set Z' = a/i(Z) is

32

I. TOPOLOGICAL PROPERTIES OF Cp(X) AND SIMPLEST DUALITY THEOREMS

everywhere dense in Rx and is of type G6 in R. Consequently, the set Z' fl Z is the intersection of a countable family of open everywhere dense sets in RX. However, RX has the Baire property [16]. Thus Z' n Z# 0. Fix h E Z' fl Z. Then h E Z' implies, by the definition of Z, that h = f + g for some f E Z. But h E Z C CC(X). Conclusion: g = h - f E Cp(X), i.e. CC(X) = RX. So, all realvalued functions on X are continuous. Thus X is discrete. To finish the proof of theorem 3.1 we need the following proposition. 1.3.2. Proposition. Let Y be an everywhere dense subspace of RT and 4) a nonempty compactum in Y, with X(4>, Y) < lto. Then r <

Proof. Since '7 = RT and 4) is compact, the requirement X(4, Y) < loo implies that X(4i, R') < Bo. But then there exists a countable family of standard basis sets in RT which completely characterizes the positioning of 4) in RT. The compactness of 4) now implies that r < Ro. We return to the proof of theorem 3.1. In a Cech complete space there is a nonempty compactum of countable character. Hence proposition 3.2 implies that if RX contains a Cech complete everywhere dense space, then X is countable. The theorem has been proved.

1.3.3. Corollary. A space CC(X) is Cech complete if and only if X is discrete and countable.

As regards the Baire property of Cp(X) the situation is somewhat different from that of Cech completeness.

1.3.4. Theorem [110], [155].

If Cp(X) is a space with the Baire property, then

every bounded set in X is finite.

Proof. Suppose there exists an infinite bounded set A C X. Put Gi = If E Cp(X):

there is an x E A such that f (x) > i}, i E N+. We show that Gi is open and everywhere dense in CC(X). Let f E Gi. Take an x E A such that f (x) > i. Put e = f (x) - i > 0 and consider the standard open set W (f, x, e) in Cp(X ). Clearly, f E W (f, x, e) C Gi. Hence G; is open in Cp(X ). Let W (g, xl, ... , xk, c') be an arbitrary standard open set in Cp(X ). Since A is an infinite set, there is a y E A \ {xl,... , xk}. There is an f E Cp(X) such that f (y) > i and If (xi) - g(xi) I < e for all i = 1,... , k. Then f E C. fl W (g, xl, ... , Xk) e). Hence G; = Cp(X). We show that fl{Gi: i E N+} = 0. Assume this to be not true,

and let f E fl{Gi: i E N+}. Then for each i E N+ there is an xi E A such that f (xi) > i. Thus f is unbounded on A, contradicting the requirements. Hence fl{Gi: i E N+} = 0, which is impossible since Cp(X) has the Baire property.

1.3.5. Corollary. If X is a k-space 1661 and CC(X) has the Baire property, then X is discrete.

4. THE LINDELOF NUMBER OF A SPACE Cp(X), AND ASANOV'S THEOREM

33

1.3.6. Corollary.

If X is a space of pointwise countable type and Cp(X) has the Baire property, then X is discrete. Since the product of an arbitrary set of separable spaces that are metrizable by a complete metric has the Baire property [66], the space RT also has the Baire property, i.e. if X is discrete, then Cp(X) is a space with the Baire property. Proposition 3.2 allows us to obtain another, somewhat curious, conclusion about properties of compactness type in Cp(X). In particular, it is clarified when Cp(X) is a feathered space.

1.3.7. Theorem. For an arbitrary space X the following are equivalent: a) Cp(X) is a feathered space; b) Cp(X) is a space of pointwise countable type; c) Cp(X) contains a nonempty compactum of countable character in CC(X); d) X is countable; e) Cp(X) is a space with a countable base.

Proof. Clearly,

[16], [66]. Since Cp(X) = RX, proposition 3.2 implies Clearly d)=,ee) (see theorem 1.1) and d)a) [16]. D. Lutzer and R. McCoy- have shown in [110] that if we adjoin to the discrete space N a point C from the remainder ON \ N of the Stone-Cech compactification of N, then Ave obtain a space Y = N U {C) for which Cp(Y) has the Baire property. V. V. Tkachuk has given an example of a space X in which every bounded set is finite although CC(X) does not have the Baire property. E. G. Pytkeev 1491 and V. V. Tkacliuk [55] have characterized those spaces X for which Cp(X) has the Baire property. This allowed V. V. Tkachuk to prove that if {X a: a E Al is a family of spaces such that all Cp(Xa) have the Baire property, then the product R{Cp(X,): a E Al also has the Baire property.

4. The Lindelof number of a space Cp(X), and Asanov's theorem In this section we establish some of the `coarser' restrictions on cardinal invariants of a space X which follow from the assumption that Cp(X) is a Lindelof space.

1.4.1. Theorem (M. 0. Asanov [18]).

For every space X and any n E N+,

t(X11) <1(Cp(X)).

Proof. Let l(Cp(X)) < r. We fix an n E N+ and show that t(X") < T. Suppose that x E 7 C X", where x= x") E X". Choose open sets U1,... , U" in X such that

ifxi=x3,then U1=Uj; (*)

ifxi0xj,then Uif1U1=0; and, finally,xiEU,for i=1,...,n.

1. TOPOLOGICAL PROPERTIES OF Cp(X) AND SIMPLEST DUALITY THEOREMS

34

Then U = U1 x . x Un is a neighborhood of x in X". We may assume that A C U, since x E A fl U.

Consider the set 4; = f f E Cp(X): f (xi) = 1, i = 1,... , n}. Clearly, 4i is closed

in Cp(X), so 1(4') < r. Put Vy = {g E Cp(X): g(yi) > 0, i = 1,... n} for all E A. Take an arbitrary f E 4?. Since x E A, f is continuous, y = (y1 i ... , and f (xi) = 1 for all i = 1'... , n, there is a point y = (y1, ... , E A such that f (Vi) > 0 for all i = 1,... , n. We have thus shown that U{ Vy: y E A} J 4i. Since 1(4i) <,r, there is a B C A such that IBI :5,r and 4) C U{Vy: y E B}. We show that x E B. Suppose x 0 B. Then there is a function fo E C9(X) for which fo E 4? and fo(X \ U" 1U,) = {0}, where the U, are open sets in X satisfying (*) and Ui C Ui,

(U1x...xUn)f1B=0.

Since fo E Vy, for some y' E B, we have fo(yi') > 0, i = 1,... , n, where y' is the ith coordinate of the point y'. Since y' E A C U, for each i = 1,... , n we have y,' E Ui. Thus y' E U;, since fo(y,') > 0 and Ui is disjoint from UU if xi # xi. Therefore x Un) fl B; a contradiction. y' E (U; x

It appears that at the moment we are very far from giving an `intrinsic' characterization of the spaces X for which CC(X) is Lindelof. Therefore it is expedient to search for criteria in order that Cp(X) be Lindelof, when X is restricted to lie in some class of spaces.

In relation to this it is useful to have in mind the following simple result.

1.4.2. Proposition.

If X is a normal space and C,,(X) is Lindelof, then every

closed discrete subspace Y of X is countable.

Proof. Since X is normal, Cp(X) is continuously mapped onto CC(Y) by the restriction map. Hence CC(Y) is a Lindelof space. But Y is discrete, thus Cp(Y) = Ri'. It remains to note that if Y is not countable, then R'' is not Lindelof. We extract the simple fact on which the previous reasoning rests.

1.4.3. Proposition.

If a space Y C X is C-embedded in X (i.e. every realvalued continuous function on Y can be extended to a realvalued continuous function on X) and Cp(X) is a Lindelof space, then CC(Y) is also a Lindelof space.

In the case of arbitrary Tikhonov spaces there is the following version of assertion 4.2.

1.4.4. Proposition.

If Cp(X) is a Lindelof space, then every discrete family of open sets in X is countable.

Proof. Let ry = {Ua: a E A} be a discrete family of nonempty open sets in X. For each a E A we choose an X. E Ua, and consider the set Y = {x: a E A). We show that Y is C-embedded in X; then, by 1.4.3, I7I = IYI < 2Io. For a fixed function f E C, (Y) and each a E A there is a ga E Cp(X) such that ga(xa) = f(xa) and ga(X \ U,) = {0}. It is easily verified that that function g = F-{g: a E A} is

4. THE LINDEL(UF NUMBER OF A SPACE Cp(X), AND ASANOV'S THEOREM

35

continuous on X, and gay = I. A space X is called a o-space if it contains a network that can be partitioned into a countable set of discrete (in X) families of sets.

1.4.5. Theorem.

If X is a normal or-space and CC(X) is Lindelof, then X has a

countable network.

In fact, by proposition 4.2 every o-discrete network in X is countable. 1.4.6. Corollary. A normal Moore space X for which Cp(X) is Lindelof, is metrizable and separable. This follows from theorem 4.5, since every Moore space has a o-discrete network.

1.4.7. Corollary.

If X is a metric space such that CC(X) is Lindelof, then X is

separable.

1.4.8. Problem. Is it true that every Moore space X for which C,,(X) is Lindeltif, is metrizable? Is this true if in addition X is assumed to be separable?

1.4.9. Problem. Let X be a space with a uniform base such that CC(X) is Lindelof. Is it then true that X is metrizable?

1.4.10. Proposition. Let Z C C ',(X), Z a Lindelof space, Y C X, and f: Y

R

a function satisfying the condition a) for each countable set A C Y there is a function g E Z such that gIA = f IA.

Then there is a function j E Z such that f IY = f . Proof. Put rl = {A C Y: A is countable}, and FA = (g E Z: 91A = f I A) for A E rl. The family f = {FA: A E i} is countably centered (i.e. if A C C and JAI < 3to, then nA 4 0) by assumption a), and all FA are closed in Z. Since Z is Lindeltif, of 0. Clearly, the function looked for is a function f E nC. The following three results are, in essence, due to A. V. Korovin.

I.4.11. Theorem. Let C,,(X) be a Lindelof space, Y C X, and let every countable set A C Y be C-embedded in X. Then Y is also C-embedded in X.

Proof. An arbitrary continuous function f : Y -+ R satisfies condition a) in proposition 4.10, where Z = CC(X). It remains to apply proposition 4.10.

1.4.12. Corollary.

Let C9(X) be a Lindelof space, and let every countable closed discrete subspace of the space X be C-embedded in X. Then e(X) < tto, i.e. every closed discrete subspace of X is countable.

36

1. TOPOLOGICAL PROPERTIES OF Cp(X) AND SIMPLEST DUALITY THEOREMS

Proof. Theorem 4.11 implies that every closed discrete subspace Y of X is Cembedded in X. We conclude, as in the proof of 4.2, that Y is countable.

1.4.13. Corollary.

Assume that CC(X) is a Lindelof space and that the space X

satisfies the condition:

/3) for two arbitrary countable sets A and B in X such that A n function g E Cp(X) for which g(A) C {0} and g(B) C {1}. Then X is a normal space.

= 0 there is a

Proof. Let A1, B, be two arbitrary disjoint closed sets in X. Put Y = Al U B1 and define the function f : Y -+ R by: f (x) = 0 if a E A1, and f (x) = 1 if x E B1. Condition /3) implies that f satisfies condition a) in proposition 4.10, where Z = Cp(X). It remains to apply proposition 4.10. Recall that a space X is called an No-bounded space if the closure of any countable set in X is compact. We have:

1.4.14. Corollary.

If C,,(X) is a Lindelof space and X is Ko-hounded, then X is

normal.

Proof. In fact, for two arbitrary disjoint compacts F and 4i in the Tikhonov space X there is a function f E Cp(X) such that f (F) C {0} and f (4i) C {1}. We now apply corollary 4.13. Obvious analogs of the assertions 4.10-4.14 hold for the space Cp(X) of bounded continuous functions and C°-embeddability.

In relation to propositions 4.2, 4.3, 4.4, and 4.12 it is appropriate to introduce the following cardinal invariant. The R-extent ef(X) of a space X is the smallest infinite cardinal r such that the cardinality of every discrete subspace A in X that is C-embedded in X does not exceed T. The following is obvious:

1.4.15. Proposition. If CC(X) is a Lindelof space, then eR(X) < N°.

5. Normality, collectionwise normality, paracompactness, and the extent of C"(X) In §4 we have seen that assumptions on the countability of the Lindelof number of Cp(X) imply essential restrictions on the space X. What conclusions can be made concerning the properties of X if CC(X) is assumed to be normal? What if CC(X) is assumed to be collectionwise normal or paracompact? In considering these questions, we come across the phenomenon that certain topological properties which are different in the class of all Tikhonov spaces coincide for the spaces Cp(X). Some results of this kind have already been given: so, by theorem 1.1,

the character and the weight of C,,(X) coincide; in particular, CC(X) is metrizable if and only if it satisfies the first axiom of countability. Note that the latter peculiarity is shared by all topological groups. The Em result of this kind is not complicate .

5. NORMALITY, PARACOMPACTNESS, AND THE EXTENT OF Cp(X)

37

1.5.1. Theorem. A space CC(X) is paracompact if and only if it is Lindelof. Proof. The Suslin number of Cp(X) is countable (see chapt. 0). It remains to refer to the theorem that a paracompactum with countable Suslin number is Lindelof (16, p. 300, no. 133).

1.5.2. Example.

We exhibit a space X for which Cp(X) is collectionwise normal but not paracompact. Let X = L(r) be the one-point Lindelofication of the discrete space of cardinality r > 11o. Then Cp(X) = ERT is the E-product of r copies of R. By Corson's theorem ([87), see also theorem 5.12 below), the space ERT is collectionwise normal. Clearly, every countable subset is closed in L(7-). Moreover, the tightness of

L(T) is uncountable. Theorem 4.1 now implies that the space Cp(X) = ERT is not Lindelof. Hence, by theorem 5.1, it is not paracompact.

We now consider the conclusions that can be made if Cc(X) is assumed to be normal. The following results (assertions T.5.3-1.5.7) are due to V. V. Tkachuk.

I.5.3. Theorem. If the space Cc(X) is normal, then it is countably paracompact. Theorem 5.3 follows from assertions 5.4 and 5.5.

1.5.4. Theorem. The space CC(X) is homeomorphic to the product Y X R, where Y = If E C'(X): f (,To) = 0} is a subspace of Cp(X) and the point xo is chosen arbitrarily.

Proof. The map /i: Cp(Y)

Y x R given by: q5(g) = (g -g(xo),g(xo)) is clearly a

homeomorphism.

1.5.5. Proposition.

If the space Y x R is normal, then the space Y x R x R is

also normal.

Proof. Let Y x R be normal. Then Y is countably paracompact. Since I x I is a metrizable compactum (where I = 10, 1) is the unit interval), the product Y x (I x I) is also countably paracompact and normal. The space Y x R x R is homeomorphic to the union of a countable family of closed subspaces of Y x (I x I). Hence Y x R x R is normal (66).

1.5.6. Theorem. The space Cp(X) is hereditarily normal if and only if it is perfectly normal.

This theorem follows from proposition 5.4 and the following assertion.

1.5.7. Proposition. normal.

If the space Y x R is hereditarily normal, then it is perfectly

38

I. TOPOLOGICAL PROPEItFIES OF Cp(X) AND SIMPLEST DUALITY THEOREMS

Proof. If a product Z x Y is hereditarily normal, then either Z is perfectly normal or all countable sets in T are closed in T (see, e.g., (16, chapt. III, no. 23)). Putting Z = Y and T = R, we conclude that Y is perfectly normal. But then the product of Y and any space with a countable base is perfectly normal [16]. Hence Y x R. is a perfectly normal space. An outstanding result concerning normality of CC(X) has been obtained by E. A. Reznichenko.

1.5.8. Theorem. Let V be a convex everywhere dense subspace of I. If V is normal, then V is collectionwise normal. We will split up the proof of this theorem in several lemmas and propositions. 1.5.9. Lemma. Let X be a convex subspace of some locally convex linear topological

space L (over R), H = {ha: a < w1} C X, 0 < t < 1, and T = {tha + (1 - t)hp: If H n T = 0, then H is discrete and closed in X; moreover, ha 34 hp if a

a,$<wl.

13,

Proof. If ha = h5 for certain a.6 < wl, then, clearly, ha E H n T, contradicting the requirements. If x E X and X E H \ {x}, then by the local convexity of L the point x lies also in the closure of T, i.e. X E H n T, contradicting the requirements. Hence the set H is closed and discrete in X.

1.5.10. Lemma.

Let Y be an everywhere dense subspace of the space IX and H an uncountable discrete closed subspace of Y. Then there are a subset F = If,,: a < wl } C H and a number t, 0 < t < 1, such that H n T = 0, where

T = {tha+(1-t)hp: a,/3<wl, a0,0}. Proof. The sets W (f, e, K) = {g E Y: Ig(x) - f (x)I < e for all x E K}, where f E Y, I > e > 0, and K is a finite set, K C X, form a base of Y. For each f E H there are a finite set Kf C X and an cf > 0 such that

W(f,ef,Kf)nH= {f}. We can choose an uncountable set F = f f,,.- a < wl } C H such that for all a < wl the inequality cf. > e is fulfilled, where c is some positive number. Put t = e/3 and, for all a < wl, W. = W (fa, c2/3, KJe ) The number t is the number we looked for. Clearly, IV,, ? fa. Hence it suffices to verify that W. nT = 0 for all T={tfa+(1-t)fa:

-.;._-

or d

12

<

it m t - : t be prns

that

3.

< _,. thet !.F.: - 1 - 0!-

5. NORMALITY, PARACOMPACTNESS, AND THE EXTENT OF CP(X)

39

Case 1. Let a 0 y. Then f7 0 W(f. e, Kf ), and thus there exists an x" E Kf, for which I ff(x`) - f7(x*)I > e. We have

I tfo(x`) + (1- t)f7(x') - f,(x`)I = I t(fa(x`) - f7(x')) + (f7(x') - fa(x'))I I f. (x*) - f7(x`)I - II fa(x) - f7(x*)I 2

3 2> f > 3 This implies that tfo + (1 - t) f7 W0. Case 2. Let a = y. Then /i 0 a and fo 4 W (f, e, Kf,). Hence there is an i E Kf, for which I fa(x) - fa(f)I > e. We have

Itfa(x) + (1- t)ff(x) - fQ(x)I = I tfa(±) - tff(i)I > t ' e = 3 '1'lcis inlplics that If,, -I- (I - t) f0 Ll Q. We now need the following simple

1.5.11. Lemma.

a. If the extent of a normal space X is countable, then X is collectionwise normal. b. If X is collectionwise normal and c(X) < No, then the extent of X is countable.

Proof a. If the extent of X is countable, then every discrete family y of nonempty closed sets in X is countable. Using the normality of X, we can separate the elements of y by disjoint open sets, and then turn to a discrete system of neighborhoods of the elements of y. Hence X is collectionwise normal. b. Let c(X) < No and X be collectionwise normal. If the set M = {xa: a E A) is discrete in X, then there is a discrete family {Ox,,: a E Al of neighborhoods of these points. Since c(X) < No, we have IAI < No and hence the set M is countable. So, e(X) < No.

Proof of theorem 5.8. Since V is everywhere dense in F, we have c(V) < lto. Hence by lemma 5.11 it suffices t prove that e(V) < 1 o. Assume the contrary, and fix an uncountable closed discrete subspace F in V. By lemma 5.10 there are a subset H = { fa: a < wI} C F and a number t, 0 < t < 1, such that H fl T = 0, where T = It f. + (1 -- t) fa : a,,6 < wj, a: ,0}.

Since H is closed in V and V is normal, there is a continuous function g: V --+ I for which g(H) fl g(T) = 0. By the factorization lemma (0.2.3), there is a countable set. A C X such that lrA(H) fl irA(T) = 0. Put h. = irA(f,), H o = {h.: a < wI}, and T o = {tha+(1-t)ha: a,/i < WI, a # /i}. Clearly, Ho = lrA(H), To = lrA(T), and, by the above, HonTo = 0. By lemma 5.9, Ho is an uncountable closed discrete subspace in lrA(V). This contradicts the fact that aA(V) is a separable metrizable space. Thus theorem 5.8 has been proved.

I. TOPOLOGICAL PROPERTIES OF Cp(X) AND SIMPLEST DUALITY THEOREMS

0

' Using theorem 5.8 it is easy to show that

.5.12. Theorem. If the space CC(X) is normal, then it is collectionwise normal. 'roof. CC(X) is clearly homeomorphic to the space Cp(X, (0, 1)), while the latter is convex everywhere dense subspace of IX. It remains to apply theorem 5.8. It is clear from the proof of theorem 5.8 that the requirement that V be a convex ubspace of IX can be replaced by the weaker requirement that for two arbitrary ,oints f, g in V the segment S(f,g) joining f and g in IX intersects with V along a et which is everywhere dense in S(f , g):

S(f,g)nV =S(f,g) subspace V C IX is called 0-convex if it satisfies this last condition. Thus the allowing generalization of theorem 5.8 holds.

.5.13. Theorem. If V is an 0-convr_c everywhere densc subspacr of 1-Y, then V s normal if and only if V is collectionwise normal. To obtain a proof of theorem 5.13 from the proof of theorem 5.8, instead of a ingle t we have to take for each a and 6 a t3 within some interval, e.g. such that /4 < tom, < e/3 (in lemma 5.9). Since the complement of a zero-dimensional set on an interval is everywhere dense, re have

.5.14. Proposition.

Let V be a convex subspace (of IX) and Y C V, Y zeroimensional. Then the set V' = V \ Y is 0-convex. Using proposition 5.14, theorem 5.13 implies

.5.15. Corollary.

Let A C Cp(X), A countable, and let the space CC(X) \ A be ormal. Then Cp(X) \ A is collectionwise normal. Theorem 5.13 does not allow us to answer the following question.

.5.16. Problem. Let the space Cp(X,D) be normal, where V = {0, 1} is discrete. I it true that Cp(X,D) is collectionwise normal? Another approach to the study of conditions of normality is related to a very general ieorem of Corson, a generalization of which is given below.

.5.17. Theorem.

Let X = fj{X,: a E Al be the product of separable metric

oaces, Y C X, Y everywhere dense in X, and let the space Z be a continuous image

fY

.If Z x Z is normal. then Z is collection wise normal.

5. NORMALITY, PARACOMPACTNESS, AND THE EXTENT OF Cp(X)

41

Proof. By lemma 5.11a) it suffices to prove that the extent of Z is countable. Assume

the contrary. Then Z contains an uncountable closed discrete subspace Y = {za: a < w1}. The sets A = {(za,xa): a < w1} and B = {(za,zp): a, l3 < wl, a # i3} are closed in the normal space Z x_Z_and are disjoint. Hence there is a continuous function h: Z x Z R such that h(A) n )T(B) = 0. Fix a continuous map 0 from Y onto Z and choose for each a < w, a point ya E Y such that Qf(ya) = za. For the sets

P={(ya,ya) a<w,}

and

T={(ya,yp): a,6<w1ia:

t3}

we have (0 x 0) (j A and (c) (T) = B. The function f = h o jb: Y x Y - R is continuous, and f (P) n f (T) = h.(A)nh(B) = 0. The space Y x Y is everywhere dense in the product fI{Xa: a E A} x rI{Xa: a E -A} of separable metric spaces, and the sets P and T are R-separated in Y x Y. By the factorization lemma (0.2.3) there is a

countable set M C A such that 't/(P)fl(T) = 0, wheret = lrMxaM: YxY -+ fI{Xa: a E M} x fI{Xa: a E M} and 7rM: Y --i fI{Xa: a E M} is the natural projection. Put s,, = ref (?/n) and P0 = {(s,t,st.,): a < wl}, To = {(4n,.,3): a, fl < wl, a fn'} arl y, 1;) and '/('j V,(T). Thus T-10 n To = 0. The set, S - Is,,: a < WI) has a limit point s' in the subspace ir,tij(Y) of the space fI{ Xa: a E All, since Al is countable, S C lrM(Y), and irM(Y) has a countable base.

Let Os' be any neighborhood of s' in lrM(Y). We find a' t3' < w1i a' # Q', for which sa,, sp, E Os'. Then Os' x Os' D (se, sa,) and Os' x Os' E) (se', spy); moreover, (sa,, say) E Po and (80r, ep-) E To. Hence (Os' x Os') n Po # 0, (Os' x Os') n To # 0, i.e. Po n To E) (s', s')-a contradiction. Since CC(X) is everywhere dense in RX (which is the product of separable metric spaces), theorem 5.17 implies

1.5.18. Corollary (Corson).

Let Y be an everywhere dense subspace of CC(X), and let Y x Y be a normal space. Then the space Y is collectionwise normal.

Yet another approach to the question when a normal space is collectionwise normal is related to elementary computations with cardinal invariants. Here it is possible to transcend the scope of theorems 5.8 and 5.17; however, we must invoke the continuum hypothesis.

1.5.19. Proposition. Let X be a normal space with countable Suslin number, and let X(X) < 00, i.e. the space X has at every point a countable base of cardinality < 2"0. Then X does not contain a closed discrete subspace of cardinality 2"0.

Proof. Assume the contrary, and fix a closed discrete subspace Y in X such that IYI = 2"0. For each point y E Y we choose a base S. of X at y such that 113,1 :5 2"0. Since X is normal, for each A C Y there is an open set UA in X such that A C UA

and UA n (Y \ A) = 0. We can extract from the family A = {V E U{lay: y E A}: V C UA} a countable subfamily 'YA such that USA C lTyn, since X has countable Suslin number. Then DyA n Y = A, and hence, if A,, A2 C Y and AI A2, then YA1

'7A2. The family yA is a countable subfamily of the family C = U{By: y E Y};

I. TOPOLOGICAL PROPERTIES OF Cp(X) AND SIMPLEST DUALITY THEOREMS

42

moreover, the cardinality of l; does not exceed FYI X(Y) = 2"0. The cardinality of the family Exp"0 a of all countable subsets of the set C also does not exceed 2"0. Since the correspondence Y J A -+ ryA C injectively maps the set Exp Y into Exp"0 t;, we conclude that ( Exp YJ < 1 ExpN° C1 < 2"0, contradicting the fact that

IExpYl> IYI=2"°. 1.5.20. Corollary.

Assume 2"0 = l it and let X be a normal space with countable Suslin number such that X(X) < 2"0. Then the extent of X is countable. We apply the results obtained to the spaces CC(X). The Suslin number of Cp(X) is always countable. By theorem 1.1, x(Cp(X)) = lXi. By invoking lemma 5.11 we now obtain from corollary 5.20 the following conclusion.

Assume 2"0 = Ri. If the cardinality of the space X does 1.5.21. Theorem. not exceed 2"0, and Y is a normal cvcrijwheiv dense subspace of (,',,(X), lhcn Y is collectionwise normal.

The assumption 2"0 = N, in 5.20 and 5.21 can be weakened to: 2"0 < 2"1. For this we have to formulate proposition 5.19 somewhat differently. A metacompact collectionwise normal space is paracompact [16]. Hence theorem 5.12 implies

1.5.22. Corollary.

If Cp(X) is normal and metacompact, then it is a Lindelof

space.

Invoking Asanov's theorem, we obtain

1.5.23. Corollary. If the tightness of a space X is uncountable, then Cp(X) cannot be simultaneously normal and metacompact.

E.g., for z > No the space Cp(LT) is normal, but not metacompact. In relation to the results obtained in this section, the following problems arise.

1.5.24. Problem. Assume that the space Cp(X) is normal. Is it true that CC(X) X Cp(X) is normal?

1.5.25. Problem.

Let Y be an everywhere dense subspace of a product 1 {Xa: a E A} of separable metrizable spaces. Is it true that Y normal implies that Y is collectionwise normal? I

'If JAI < N,, then the answer is'yes' (D. P. Baturov). If c = 2'° = 2K', then Ic contains an everywhere dense normal subspace which is not collectionwise normal (D. P. Baturov)

6. THE BEHAVIOR OF NORMALITY UNDER THE RESTRICTION MAP

43

6. The behavior of normality under the restriction map between function spaces Let CC(X) be normal and Y C X. Using the restriction f

fly, the space

CC(X) is mapped onto the subspace C,(YIX) of the space C,(Y). The properties of the restriction map Try: Cp(X) --+ CC(YIX) do not give reasons to expect that the space CC(YIX) will be normal also: if Y is not closed in X, the map Try need not even be open. If Y is closed in X, then Try: Cp(X) -+ C'(YIX) is a continuous open neap. However, in general even under such maps normality need not be preserved: by a theorem of V. I. Ponomarev [1.6], (66], every space with the first axiom of countability is the image of a metrizable spare under a continuous open map. Hence V. V. Uspenskii's theorem on the preservation of normality under a restriction map, which we will prove

in this section, is a quite unexpected result. We need the following version of the factorization lemma.

1.6.1. Lemma.

Let Y he an everywhere dense subspace of a product X = ff (X,,:

(e E A} of sepan(.ble. 116Ct1 i : 3ltllces X,,, and let I', 7' C Y. Then the following conditions

are equivalent:

a) there are open sets U and V in X such that P C U, T C V, and U n V = 0; b) there is a countable set M C A such that the sets TrM (7') and 1rAf (P) are separated in the space 1rM(Y C TrM(X) = fJ{Xa: a E Al, i.e. Try((P) n 7rm(T)=0=TrM(P)nlrM(T).

Proof a) .b). Let a) be fulfilled, and let U and V be sets as in a). Take open sets U, Vin Xsuch that UnY=Uand VnY= V. Then (UnV)nY=UnV=0, and since Y is everywhere dense in X, UnV = 0. By yo (respectively,'yV) we denote some maximal disjoint family of elements of the standard base of the product X, lying in U (in V). Then Iryol <_ No, I'yvI <_ No (since c(X) < l o), and 1Jryo J U, 7f, D V. By M we denote the family of all a E A on which at least one element of the families 'y;, and yV depends. The set. M is countable, and, clearly, Tr1i1TrM(Uyf,) = Uyo C U, Trey 7rM(U7V) = U-y C V. Hence, TrM TrM(UyO) nTrM TrM(Uyf,) C Un V = 0, implying that TrM(Uyu) n 7rM(Uyv) = 0. But TrM(Uyu) and TrM(UyV) are open sets in TrM(X),

and are everywhere dense in, respectively, TrM(U) and TrM(V). Since the sets 7r5(U) and Ti-M(V) are open in fI{X,,,: a E A}, the sets TrM(P) and Tryq(T) are separated in TrM(X), and thus in TrM(Y). b) .a). Using the notation of b) we have: IrM(P) UTrM(T) C 1rM(Y) \ (TrM(P) U TrM(T)). Put Z = 1rM(Y) \ (TrM(P) U 7rM(T)). Since M is countable, Z has a count-

able base, and is hence normal. The sets TrM(P) and TrM(T) are closed in Z, and are disjoint. Hence Z contains disjoint open neighborhoods W and G of these sets: TrM(P) C W, 1rM(T) C G. Since Z is open in TrM(Y), the sets 1V and G are open in TrM (Y), so that the sets Trey' (W) and 7rm (G) are open in Y. Moreover, P C TrAAtV),

T c Trj'(G).

1.6.2. Theorem [17].

Let Cc(X) be normal, and let Y be closed in X. Then the space C,(YIX) = 1ry(Cp(X)) is normal also.

I. TOPOLOGICAL PROPERTIES OF Cp(X) AND SIMPLEST DUALITY THEOREMS

44

Proof. The space CP(YIX) is everywhere dense in R. Let P, Q be disjoint closed sets in Cp(YIX), and P = irY'(P), Q' = iry'(Q). It suffices to find a countable Z C Y such that 7rz(P) and 7rz(Q) are separated in irz(Cp(X)). The space CC(X) is normal.

Hence there is a countable Z' C X such that irz.(P) and irz'(Q') are separated in irz.(CC(X)). We show that without loss of generality the set Z' can be replaced by Z = Z'f1Y, i.e. that irz(P) = P and 7rz(Q') = Q are separated. Assume the contrary. Let, e.g., 7rz(P) fl c1T(lrz(Q')) = 0, where T = irz(Cp(X)). We choose f E P' such that f Iz E 7rz(Q'), and show that f (z' E clr(7rz'(Q')), where T' = rrz'(Cp(X)). Let K C Q' be a given finite set, and let e > 0. Put Kl = KnY, K2 = Kfl (X \Y). Since f 1z E irz (Q), there is a g E Q such that Jg(x) - f (x) I< e for all x E K1. There is a g' E Q' for which lry(g') = g. Since Y is closed in X, we can choose an h' E CC(X) such that h'ly 0 and h'(x) = f (x) - q'(x) for x E K2. Let now h = h' + g'. Then

hey = g (and thus h E Q'), and ih(x) - f (x)l < e for all x E K. Since K and e are arbitrary, this means that f Iz' E az'(Q'). Thus, irz'(P') and az'(Q') cannot be separated, a contradiction. Hence 7rz(P) and irz(Q) can be separated, which was required to prove.

1.6.3. Corollary.

If X and Cp(X) are normal, then all closed discrete subsets of

X are countable.

Proof. Let A be a closed subset of X. Then Cp(A) = 7rA(Cp(X)) by the normality of X. By theorem 6.2, Cp(A) is normal. If A is discrete, then CC(A) = RA, and since RA is not normal if A is uncountable, A must be countable.

1.6.4. Corollary. If X is metrizable, then Cp(X) is normal if and only if X has a countable base.

1.6.5. Corollary.

If X is a normal Moore space and Cp(X) is normal, then X is

metrizable.

1.6.6. Corollary.

If X is a normal o -space and Cp(X) is normal, then X has a

countable network.

Note that the following theorem can be proved similarly to theorems 5.7 and 5.11.

1.6.7. Theorem. normal.

The space Cp(YIX) is normal if and only if it is colleclionwise

CHAPTER II

Duality between invariants of Lindelof number and tightness type

1. Lindelof number and tightness: the Arkhangel'skii-Pytkeev theorem The Lindelof number is one of the most important cardinal invariants of compactness type. At first sight, the relatively new invariant lightness seems unrelated to it. Note that whereas the Lindelof number is by nature a global invariant, tightness is essentially a point invariant. However, in §4 of chapt. I we have stated M. O. Asanov's theorem: If CC(X) is a Lindelof space, then the tightness of every finite power of X is countable. This theorem has no direct converse: if X is an uncountable discrete space, then X" is discrete and hence t.(X") < 1 o, but the space Cp(X) = Rx is not Lindelof. Fortunately, however, there is a `reflected' theorem, which turns out to be symmetric, i.e. a duality theorem.

11.1.1. Theorem (A. V. Arkhangel'skii-E. G. Pytkeev [1], [48]). l(X") _< r for all n E N+ if and only if t(C,(X)) < T. Proof. Sufficiency (A. V. Arkhangel'skii [1], 13]). Let l(X") < r for all n. E N+. Fix an f E CC(X) such that f E 7 C Cp(X). Put e" = 1/n. For each point

C = (xi, ... , x") E X" there is a gf E A such that Igf(xi) - f (xi) I < e" = I/n for i = 1,... , n. Since gf and f are continuous, there is a neighborhood O(xi) of xi in X such that jgf(y) - f (y)I < 1/n for all y E O(xi). Consider the neighborhood Uf, = O(xi) x x O(x") of C in X". Then 11" = {Uf: E X"} is a cover of X. Let pn C rt" be a cover of X" such that 1%'I r. Take B" = {gf: Uf E i }, and consider B U{B": n E N+}. It is clear that B C A and JBI < T. It remains to prove that f E Take a collection y',.. . , y" and an c > 0. We may assume (by diminishing the collection, if necessary) that 1/n < e. There is a C E X" for which (y,. .. , y") E Uf. Then Igf(yi) - .f (yi)) < 1/n < e. Sufficiency (E. G. Pytkeev [48]). Let t(Cp(X)) < r, n E N+, and let -y be an open cover of the space X". A finite system µ of open sets in X is called small relative to y (-y-small) if for any Vi, ... , V" E p there is a G E ry such that V x ... x V" C G. We denote by E the family of all (finite) -y-small families of open sets in X. For µ E E, let A. = f f E C,,(X): f (X \ Uµ) = {0}}. We put A = U{A,: p E E} and show that 45

11. DUALITY BETWEEN INVARIANTS OF LINDELOF NUMBER AND TIGHTNESS TYPE

46

A = C,(X). Let f E CC(X) and K C X. We construct in an obvious manner a finite family OK of open sets in X such that for any collection (yl,... , y") E IC' there are V1, ... , V,, E 0K

and aGEryfor which y;EViand VI

xK

E V} and consider the family ILK = {W=: x E K}. E Clearly, K C UµK. The family µK is y-small. Indeed, consider a set of the form FVi, x x W. There are V1,...,V"E OK such that xieVi and VI Since W, C Vi for

i=1,...,n,wehave

Take now a function g E CC(X) such that 91K = f lK and g(X \ UPK) = {0}. This

is possible, because K C UµK. Clearly, g E A,,, C A, and g lies in all standard neighborhoods of f based on the set K. Hence f E A. In particular, take for f the function f1 E C,(Y) identically equal to 1. By the above, f1 E A, and since t(CD(X)) < T there is a B C A such that IBS < T and f1 E B. Then there is a subfamily Eo C E for which B C U{A,,: tt C-.6} and I£I < T. Let p E Eo. For each f = (V , ... , V,<) E µ" we fix a GE E y such that V x x V. C C. Put yi, = {Ge: 1; E p"}. The family y,, is finite, hence the cardinality of the family y = U{-yµ;µ E Eo} does not exceed T. We show that y covers X". Let (x,, ... , x") E X" and U = f f E C9(X ): f (xi) > 0

for i = 1,...,n}. The set U contains f1 and is open in CC(X). Since fi E B and B C U{A,,: p ED}, there is a p0 E Eo such that U fl A,,O # 0. For g E AµO we have g(xi) > 0 for i = 1, ... , n and g(x) = 0 for x. Upo. Thus xi E Ulto for all i = 1, ... , n. Take V E µo, i = 1, ... , n, such that xi E Vi. Then (x1, ... , x") E VI x x V" C Gt E yµ0 Cy. The theorem has been proved [48]. For an arbitrary cardinal invariant q5 we put q5'(X) = sup{0(X"): n E N+} (to be more precise, we should have written (O)*, not q''). Theorem 1.1 can now be rephrased as

11.1.2. Corollary.

The cardinal invariant l' is supertopological; in fact: if the spaces CC(X) and Cp(Y) are homeomorphic, then l'(X) = l'(Y).

If 1'(X) < T, then for the space X x N+, which is the free sum of ko copies of X, we obviously have 1'(X x N+) < T. But the space Cy(X x N+) is (canonically) homeomorphic to the space (Cp(X))"0, which is also obvious. Finally, (CC(X))"° is (also canonically) homeomorphic to the space Cp(X, R"0). Hence theorem 1.1 implies

11.1.3. Corollary. If 1'(X) < 11o, then t(CC(X, R"°)) < Ito and the tightness of the countable power of CC(X) is also countable.

This assertion can be trivially generalized to the case of an arbitrary cardinal T. In particular, theorem 1.1 can be applied to compacta.

11.1.4. Corollary. If X is a compactum, then t(Cp(X)) < No.

1. LINDELOF NUMBER AND TIGHTNESS: THE ARKHANCEI,SKIT-PYTKEEV THEOREM

47

A space is called a P-space if every subset of type C6 is open in it. If X is a Lindelof P-space, then l(X") < 1 o for all n E N+. Therefore,

II.1.5. Corollary. If X is a Lindelof P-space, then t(CC(X)) < 8o A Lindelof feathered space can be defined as a space which can be perfectly mapped

onto a space with a countable base. The images of Lindelof feathered spaces tinder continuous maps are called Lindelof E-spaces 11231. The product of countably many Lindelof E-spaces is Lindelof (123].

11.1.6. Corollary. I f X is a Lindelof E-space, then t(CC(X)) < 1 o Even when X is a compactum satisfying the first axiom of countability, it is not possible to `reflect' theorem I.1. We now give an example of a perfectly normal compact.um X such that C1,(X) is not only not Lindelof, but does even contain a discrete subspace of cardinality 2"0.

11.1.7. Example.

For X we take the well-known `two arrows' compactwn 116J. The set X = 10,1) x {0, 1} is endowed with the topology generated by the natural lexicographic ordering. For each real number a such that 0 < a < 1, we take f. E Cp(X) such that fa(x) = 0 if x< (a, 0) and ,,ax) = 1 if x > (a, 1). We show that the set A = {fa: a e (0, 1)} is discrete and closed in Cp(X). The discreteness of A follows from the fact that W (f., (a, 0), (a,1),1/2) fl A = { fa}. The definition of the topology of pointwise convergence readily implies that the only limit points of A in Rx are the functions gQ and g.+, where gd (x) = 1 for x < (a, 0), ga (x) = 0 for x > (a, 0), and g,+(x) = 1 for x < (a, l) and qa (x) = 0 for x < (a,1). Since gQ , g,+ V C9(X), we find that A is closed and discrete in CC(X). The space (Cp(X))Ho is canonically homeomorphic to the space Cp(X x N), where N = {0, 1, 2,... } is discrete. Clearly, if X" is Lindelof for all n E N+, then the space Y = X x N is Lindelof. Thus, theorem 1.1 implies

II.1.8. Corollary.

If the tightness of Cp(X) is countable, then the tightness of

(Cp(X))N0 is also countable.

On the other hand, T. Pshimusin'skii has constructed spaces X and Y such that X" and Y° are Lindelof for all n E N+, but X x Y is not Lindelof. For the free sum X (D Y of the spaces X and Y we then have: Cp(X fl) Y) is canonically homeomorphic

to the product Cp(X) x C9(Y), and by theorem 1.1, t(CC(X) x CC(Y)) > fto, while t(Cp(X)) < No and t(Cp(Y)) <_ No. In relation to theorem 1.1 and corollary 1.2 the following questions arise naturally:

11.1.9. Problem. Does there exist a natural topological property of Cp(X) which would characterize whether the space X is Lindelof?

48

II. DUALITY BETWEEN INVARIANTS OF LINDELOF NUMBER AND TIGHTNESS TYPE

Is the Lindelof number preserved under t-equivalence? In 11.1.10. Problem. particular, is it true that a space which is t-equivalent to a Lindelof space is itself a Lindelof space?

II.1.11. Problem.

Let the space CC(X) be Lindelof. Is then the tightness of C,Cp(X) countable? Is the tightness of Lp(X) countable? Since for each n E N+ the space Xn is homeomorphic to a closed subspace of the space Lp(X) (see chapt. 0), a positive answer to any of these problems will give a considerable strengthening of Asanov's theorem 1.4.1. By theorem 1.1, the first part of problem 1.11 is equivalent to

11.1.12. Problem. Let the space Cp(X) be Lindelof. Is it then true that the space Cp(X) x Cp(X) is Lindelof?

2. Hurewicz spaces and fan tightness In this section we prove a variant of theorem 1.1: a property of X stronger than the Lindelof property can be characterized by a property of CC(X) stronger than tightness. It is not excluded that there may be other theorems of similar nature. In particular, it would be extremely interesting to characterize a-compactness of X by a tightness type property of Cp(X ). Theorem 2.2 may be regarded as an approximation to this characterization. A space X is called a Hurewicz space (see [109] for a survey of its properties) if for each sequence (N: n E N+} of open covers of X there exist finite sets A. C 'yn such

that Uan covers X. All compacts and all a-compact spaces are such. The property of being a Hurewicz space is preserved under transition to a closed subspace, under continuous maps, and under taking countable unions. Any Hurewicz space is Lindelof. Among the separable metric spaces they are characterized by the possibility to extract from every base a locally finite cover [109]. Consequently, J, the space of irrational numbers, is not a Hurewicz space. Recall that a space is called analytic if it is a continuous image of J. A. V. Arkhangel'skii [15] and J. Calbrix have shown that if a Hurewicz space is analytic, then it is a-compact. We will need

11.2.1. Proposition. A space X is compact if and only if XK0 is a Hurewicz space. Proof. Let XH0 be a Hurewicz space. Then X is a Lindelof space, and it suffices to prove that X is countably compact. However, if X D A is a closed discrete subspace, then Ak0 is a closed subspace of X10 homeomorphic to J. Hence J is a Hurewicz space, a contradiction. We say that the fan tightness of X does not exceed lZo (and write vet(X) < lio) if for each point x E X and each countable system {An: n E N+} of sets in X such that x E fln4n there are finite sets Bn C An for which x E UnB,. Replacing No in the above by an arbitrary cardinal r > No, we obtain the definition

49

2. HURFWICZ SPACES AND FAN TIGHTNESS

of the cardinal invariant vet(X), the fan tightness of X. If Y C X, then vet(Y) < vet(X). Always, t(X) < vet(X) < x(X). For the Frechet-Urysohn fan V(No) we have t(V(No)) = No = IV(Ro)I < vet(V(No)). Thus, continuous closed maps can increase fan tightness.

11.2.2. Theorem. For an arbitrary space X the following are equivalent: a) X" is a Hurewicz space for any n E N+; b) vet(CC(X)) < No.

Proof. b)=a). Let n E N+ and {yk: k E N+} a countable system of open covers of X". A system p of covers of X is called yk-small if for any Vi,. .. , V" E p there is a x V C G. Denote by Ek the family of all finite yk-small G E yk such that VI x systems of open sets in X. For µ E £k we put F,, = f f E C9(X): f (X \ Up) = {0}). We show that the set Ak = U{F,,: p E £k} is everywhere dense in C,,(X). Let f E Cc(X) and K C X, K finite. There is a finite family 0 of open sets in X such that for any (y,... , y") E K" there are VI, ..., V" E 0 and a G E yk satisfying the conditions: y, E V,, and VI x x V,, C G. Clearly, K C U0. For X E K we put. W,. = n{V E 0: V x} and let. µK = {W=: x E K}. Clearly, K C UILK. The family µK is yk-small. In fact, take an arbitrary Ws, x

x Id 2,,, where x, E K.

There are V1,...,V"E0and aGEyksuch that Vi E V, and V for i = 1,.. . , n, we have that Wi,j x ... x 1,V.,, c G. Take a function g E CP(X) such that g1K = f Ix and g(X \ UAK) = {0}. Clearly, g E F,,K C Ak, and g lies in all standard neighborhoods of f based on K. Let f - I on X. By the above,

f E Ak for all k E N+. Since vet(C9(X)) < No, there are finite sets Bk C Ak for which f E E. There is a finite subfamily Pk C &L such that each function g E Bk is p-small with respect to some p E Pk. Let p E Pk. For each _ (VI, ... , V") E µ" we choose a set Gf E yk such that The family Ak = {G{: 1; E ti E Pk} is finite, since Vk is finite and every µ E Pk is finite. Clearly, Ak C yk. We show that the family UkAk covers X. µ",

T a k e a n arbitrary (x1, ... , x") E X" and put U = {f E ? ( ) : f (xi) > 0, i = 1,... ,n}. The set U is open in C,,(X), and f E U. Since f E UkBk, there is a k' E N+ i.e. there is au'-small for which U n Bk # 0. Then U n F,,..o 0 for some a* E

function gEU. Wehave g(xi)>0fori=1,...,n,andg(x)=0 for all xEX\Uµ'. x V"CG{ Take Vi EA*such that x1Ei,,i=1,...,n. Then (xl,...,x")E V I x for some GE E \k Hence (Xi, . , Xn) E U Uk yka)=b). Let X' be a Hurewicz space for all n E N+. Fix f E CC(X) and a family {Ak: k E N+} of sets in CC(X) such that f E flkAk. Fix also n E N+ and k E N+. For each T = (x1,...,x") E X" there are gg,k, k E Ak, such that Igg,k(x;) - f(x;)I < 1/n for all i = 1, ... , n. Since the functions 9--,k and f are continuous, there is a neighborhood Oi of x; x O. is a such that Igg,k(yi) - f (y;)I < 1/n for all y, E Oi. The set Vz,k = 01 x neighborhood of 7 in X". Thus, y",k = {VV,k: I E X"} covers X", and Ig2,k(yi) P Y01 < 1/n for all (y1, ... , y") E VV,k. Since X" is a Hurewicz space, there are finite

50

IL DUALITY BETWEEN INVARIANTS OF LINDELOF NUMBER AND TIGHTNESS TYPE

sets Pn,k C X' such that the family U{.\n,k: k E N+, k > n}, where'n,k = {VY,k: T E Pn,k}, covers Xn. The set Bk = {g2,k: T E Pn,k} is finite, and Bn,k C A. But then Bk = U{B,,,k: n < k} is a finite set, and Bk C Ak. We show that f E Uk Bk. Take arbitrary yl, ... , y,, E X and an e > 0. We may assume that 1/n < e. There Then (yr,... , yn) E V=,k for some is a k` > n for which (y,, ... , yn) E 2 E Pn,k . We have gs,k. E Bn,k. and I9=,k- (Yi) - .f (yi) I < 1/n < e for i = I,... , n. But Bn,k C Bk., since n < k'. Thus, 91,k E Bk and f E UkBk. The theorem has been proved.

11.2.3. Corollary. If X is an analytic space, then the conditions: a) vet(CD(X)) < ?1o, and b) X is o -compact, are equivalent.

11.2.4. Corollary. If X ... Y, and Xn are Hurewicz spaces for all n E N+, then Y" are also Hurewicz spaces for all n E N+. The space V(R0) cannot be embedded in CP(X) if the X" are Hurewicz spaces for all n E N+ (in particular, if X is o-compact).

11.2.5. Corollary.

11.2.6. Corollary. If vet(CC(X)) < lto, then also vet((Cp(X))"0) < No. In relation to corollary 11.2.5 there arise the following questions.

11.2.7. Problem. Can V(No) be embedded in CP(X) for some Hurewicz space X?

11.2.8. Problem. Let X L Y, and let X be a Hurewicz space. Is it then true that Y is a Hurewicz space?

11.2.9. Theorem. If X"0 - % Y"0, and X is a compactum, then Y is a compactum. Proof. Since X"0 is a compactum, Y110 is (by corollary 2.4) a Hurewicz space. It remains to apply proposition 2.1.

11.2.10. Theorem. CP(X) is a Hurewicz space if and only if X is finite. Proof. Let CP(X) be a Hurewicz space. We show that X is pseudocompact. Assume this to be not true. Then there is an A C X such that IAI = No and each function g: A -+ R can be extended to a realvalued continuous function on X. Thus, RA is a continuous image of the Hurewicz space CC(X), hence is a Hurewicz space. But RA is analytic and not or-compact, a contradiction. So, X is pseudocompact. If X is infinite, there is a continuous map 0: X --. R"0 for which the space q(X) C R"o is infinite. Clearly, ¢(X) is a compactum of countable weight. The pseudocompactness of X now implies that under the inverse map the space CP(4i(X )) is homeomorphically mapped onto a closed subspace in CP(X) (for ¢ is automatically R-quotient, see 1141). Thus, CD(q5(X)) is a Hurewicz space. It remains to refer to the following proposition.

3. FRI1CHET-URYSOHN PROPERTY, SEQUENTIALITY, AND THE K-PROPERTY OF Cp(X)

51

11.2.11. Proposition. If X is a compactum of countable weight, then Cp(X) is an analytic space; if, moreover, Cp(X) is a Hurewicz space, then X is finite.

Proof. The space Cp(X) is analytic as a continuous image of the separable Banach space C(X). If Cp(X) is also a Hurewicz space, then it is o-compact [15]. By assertion 1 from §2, chapt. I we conclude that X is finite.

3. Frechet-Urysohn property, sequentiality, and the k-property of CC(X) We have already clarified for which X the space Cp(X) has countable tightness. The notion of space of countable tightness arose as a natural generalization of the classical notion of sequential space. Recall that a space X is called sequential if, and only if, for each nonclosed set A in X there is a sequence in A converging to some point in X \ A. If any set in X whose intersections with all compacta in X are closed in X, is closed, then X is called a k-space [66]. The sequential spaces form an important subclass of the class of k-spaces. In its turn, the Frechet-Urysohn spaces are distinguished among the sequential spaces. They are characterized by the fact that if a point x in such a space X lies in the closure of a subset A C X, then there is a sequence in A converging to x. In this section we show that the FrechetUrysohn property, sequentiality, and the k-property, which are different in the class of all Tikhonov spaces, coincide for spaces Cp(X) (E. G. Pytkeev [47], J. Gerlits and Sz. Nagy [95]).

The most refined part of our considerations is the proof of the fact that if CC(X) is a k-space, then its tightness is countable.

We also give a criterion for C,,(X) to be a Frechet-Urysohn space in terms of the topology of X. Using this criterion it is established that if Cp(X) is a FrechetUrysohn space, then (Cp(X))No is such.

A family A of subsets of a space X is called an w-cover of this space if for each finite set KCX there is a U E A such that K C U. For w-covers the notion of being inscribed is naturally defined: A, is inscribed in A2 (written as A, > A2) if each U E A, is contained in some V E A2. The family of all w-covers of X is directed by the relation of being inscribed: for arbitrary open w-covers A,, ... , An of X there is an open w-cover A of X such that A > A for all i = 1, ... , n. In fact, we may take, e.g., for A the family { U, fl . . . fl U,,:

U, E A,, i = 1,...,n}. If (: _ {An: n r= N+} is a sequence of subsets of X, then the set lim inf = U{fl{Al: i > n}: n E N+} is called the lower limit (limes inferior) of the sequence l:. It is obvious that x E lim inf t; if and only if x belongs to all terms of the sequence t; from some index onwards. Instead of B = liminf{An: n E N+} we also write A. B. We need the following simple proposition.

11.3.1. Proposition. If lim inf e = X, then C is an w-cover of X. Let 0 be some topological property. We write X F- 0 if the space X has the property 0.

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II. DUALITY BETWEEN INVARIANTS OF LINDELOF NUMBER AND TICHTNESS TYPE

11.3.2. Theorem. The following assertions are equivalent: a) Cp(X) is a Frechet-Urysohn space; b) X has the property y: for any open w-cover I) of X there is a sequence f C I) such that Jim inf l; = X; c) X has the property yI : for any sequence 117n: n E N+} of open w-covers of X {Un: there are Un E 77n, for all n E Ni', such that lim inf e = X, where n E N+}; d) (Cp(X))Ho is a Frechet-Urysohn space. Proof. The implication d)=*,a) is obvious. We prove a)=b). So, let 77 be an w-cover of X. Put A = If E Cp(X): f-I(R\ {0}) C U for some U E i7}. Then fo E 4, where fo - I. Since a) holds, there is a sequence {fn: n E N+} C A converging to fo. For each n E N+ we take a Un E 11 for which fnl(R\ {0}) c U,,, and show that C = {Un: n E N+} satisfies b). For any x E X we find an n(x) E N+ such that f,, E W(f0, x,1) for all n > n(.T.). But this means that fn(x) > 0, i.e..T. E fn I(R\ {0}) C U for all n > n(x), i.e. X = lim inF i;. Thus X I- -y. b) .c). This implication is obvious if X is a finite space, for any w-cover of X then contains X as an element. If IXI > Ho, we fix a subset Y = {xn: n E N+} such that

xi 0 x, if i # j, for arbitrary i, j E N+. Suppose we are given an arbitrary sequence {77n: n E N+} of open w-covers of X. Without loss of generality we may assume that 77n+1 > 71n for n E N+. Put An = {U \ {xn}: U E T7n} and u = {µn: n E N+}. We show that u is an w-cover of X.

Let K C X, K finite. For each n E N+ there is a Un E ,jn such that K C Un. There Thus 7c is, moreover, an no E N+ such that K. Then K C UnO \ {xo} E is an open w-cover of X. By b) we can fix a sequence bl = {Un: n E N+} C 71 such that lim inf bi = X. The definition of it implies that Un = Vk., \ {x"}, where Vk E 77k,,, n E N+. The sequence {kn: n E N+} is not bounded. In fact, if we would have kn < in E N+

for all n, then no element of S1 would cover the set {x,. .. , x,}, although 1 is an w-cover of X by proposition 3.1. Thus, passing to a subsequence and noting that the lower limit of a subsequence equals the lower limit of the sequence, we find

that kn+1 > kn, and hence kn -b oo. We put ko = 1 and fix for each n E N+ a such that Wp E 17,, p E {k,,,.. . , kn_1 + 1}, as follows: Wk,, = Vk,,, 4Vk._I E C We now put = {Yl n: n E N+}, and show that lim inf C = X. Indeed, let x E X and n E N+ be such that x E Vk, for p > n. Then x E WI for all I > kn, since if kp < I < kp+1 i where p > n, then x E Vko+,, and thus x E W, by the construction of . c)-#-d). It is well known [3] that (Cp(X))"0 is homeomorphic to Cp(X, R"0). Fix a co E RIO, and let 8 = {On: n E N+} be a countable base at co in RIO such that On+] C On for all n E N+. Let X i- yl.

It suffices to prove that fo e CC(X, R"0) is a Frechet-Urysohn point, where fo(X) = {co}. It is easily seen that the sets of the form W (fo, K, On) = If E Cp(X, RIO): f (K) C On}, where n E N+ and K C X is finite, form a base at fo in

3. FRE`CHET-URYSOHN PROPERTY, SEQUENTIALITY, AND THE K-PROPERTY OF Cp(X)

53

CC(X, R"0). Fix an A C R"O) such that fo E'A\ A, and let ryn = { f `'(On): f E A}. Then ryn is an w-cover of X for each n E N+. In fact, if K C X is finite,

then there is an f E W (fo, K, On) fl A, and this means that K C f '(0.) Using -

ryj we choose fn E A, for all n E N+, such that lim inf l; = X, where

nEN+}. We show that the sequence [fn: n E N+} converges to fo. Let W(fo, K, On) be an arbitrary standard neighborhood of fo, where K C X is finite and n E N+. For

x E K we fix an n(x) E N+ such that x E fl{ fk(Ok): k > n(x)). Then, clearly, K C I f;-'(00: k > m}, where m, = max{n(x): x E K}. Put I = max{m,n) and take k > I arbitrarily. Then K C fk'(O,), i.e. fk(K) C Ok C On. Thus fk E W(fo, K, On) for any k > 1, and the convergence of { fn: n E N+} to fo has been proved. We write X F- 4' if for any open w-cover rl = U{77n: n E N+} of the space X, where 77n+1 D rln, there is a sequence = {Xn: n E N-F} such that rln is an w-cover of Xn

for each n E N+, and Jim inf _= X.

Remark. It is not necessary to assume that Xn is w-covered by precisely the family 77n; it suffices that Xn is w-covered by some /7k. Indeed, if X;, , X and X;, is wcovered by r1k (kn E N+, n E N+), then we may assume that kn < kn}1, ko = 0, and Xo = 0. Put Xk = Xn if kn < k < kn+1; then Xn X and Xn is w-covered by 77n. We say that a space X has the property a if we can extract from any open w-cover of X a countable w-subcover.

11.3.3. Theorem. X F- -y if and only if X F- 0 and X F- e. Proof. Let X F- -y. Proposition 3.1 implies that X F- e. Let n = U{r7n: n E N+} be X. an open w-cover of X. Property y implies that there are Gn E 17 such that Gn If G. E r7k,,, then Gn is w-covered by 77k.. Conversely, assume that X F- ¢ and X F- e, and let r1 be an open w-cover. By property a we may assume that 1771c No. Now X F- 0

implies that there is a sequence Xn C X such that Xn - X and Xn is w-covered by 17n, where 77n denotes the first n elements of r7 with respect to some enumeration. But

a finite family can be an w-cover of a set if and only if this set is contained in some element of the family. So there is a G. E r7n for which Xn C Gn. Clearly, G. --* X.

11.3.4. Theorem. Let X F- 0. Then: a) if f : X -+ Y is a continuous map from X to Y, then Y F- 0; b) if Z is closed in X, then Z F- ¢. Proof. Let n = U{i7n: n E N+) be an open w-cover of Y such that 77n C ?7n+1. Put p= U{pn: n E N+}, where Un == {f-'(U): U E r7n}. Then p is an open w-cover of X if X. -. X and Xn is w-covered by p,,; this proves a).

Property b) readily follows from the fact that Xn - X implies Xn n z -r Z, and that fact that the correspondence U --> U U (X \ Z) between open subsets of Z and X preserves the property of being an open w-cover. The theorem has been proved.

54

It. DUALITY BETWEEN INVARIANTS OF LINDELOF NUMBER AND TIGHTNESS TYPE

11.3.5. Lemma.

The space Cp([O,11) is not sequential.

Proof. Let Jr.: n E N+} be a dense set in I = [0, 1), and let t3 = {U.: n E N+} be a base of I such that p(U,,) < 1/2 (p being Lebesgue measure) for all n E N+, and such that for each finite K C I there is an n E N+ with K C U,,. For each n E N+ we choose an fn E CC(I) satisfying

k
fndx>2,

fn(I)CI.

Then Z = I fn: n E N+} U {g}, where g = 0, is the required set. In fact, if f E CC(I) is a limit point of Z, then f (rn) = 0 for all n E N+, whence f = g (since Z is closed and Z \ {g} is discrete). For gn E Z \ {g} we cannot have gn g, since the inequality

f gn dx > 1 I

n E N+,

implies by Lebesgue's dominated convergence theorem that, as n. oo, the not converge to zero on a set of positive measure. The lemma has been proved.

do

II.3.6. Corollary. If X F- 40, then ind X = 0. Proof. Let x E X and U a neighborhood of x in X. Choose f E C,(X) such that f (X) C [0, 11, f (x) = 1, and f (X \ U) = {0}. Theorem 3.3 implies that f (X) F- q5. Since w(f (X)) = 8o, we have f (X) F- e. Thus, f (X) I- -y, i.e. CI,(f (X)) is a FrechetUrysohn space. By lemma 3.5, f (X) # (0, 11, i.e. there is a 6 E (0, 1) such that 6 0 f (X). Then f-1((b,1]) is an open-closed set containing x and contained in U. Now we are ready to prove the main theorem in this section.

11.3.7. Theorem. For any space X the following conditions are equivalent: a) Cp(X) is a k-space; b) Cp(X) is sequential; c) CC(X) is a Frechet-Urysohn space.

Proof. The implications c)b)=a) are obvious. We prove that a)=c). By the above, it suffices to prove that a)= (X I- y) or, equivalently, that a) ((X I- 0) A (X I- e)).

11.3.8. Lemma. Let Cp(X) be a k-space. Then X F- q5. Proof. Assume that the assertion of the lemma is not true, i.e. CC(X) is a k-space, but X F/ 0. Then there is an open w-cover t) = U{77n: n E N+} of X refuting the

fact that X I- 0. We denote by X (f < n) the set {x E X: f (x) < n}, and put for n E N+, An _ If E Cp(X): X (f < n) is w-covered by the family T/n}, and A = U{ An:

nEN+}.

The set An is closed for each n; indeed, if f E CC(X) \ A,,, then X(f < n) is not w-covered by n,,, hence there is a finite set F C X(f < n) such that no element of 11n

3. FRECHET-URYSOHN PROPERTY, SEQUENTIALITY, AND THE K-PROPERTY OF Cp(X)

55

contains F. Then f E G = {g E_ Cc(X): for all x E F, g(x) < n} C Cc(X) \ A,,, as required.

On the other hand, the set A. is not closed in C9(X): if fo = 0, then f, E A \ A. Using a), we choose a compact set C C C9(X) such that C fl A is not closed in C. All projections from C onto the line are bounded, i.e. for any x E X there is an n(x) E N+ such that f (x) < n(s) for all f E C. Let X = {x E X: n(x) < n}. Then n E N+} = X, and Xit1 D X,,. Thus X. - X. Because the family 11 refutes X I- 0, there is an m E N+ such that no 77k is an w-cover of Xm.

But C n A. = 0 if n > m. In fact, let f E A,,, where n > m.. Then X (f < n)

is

w-covered by the family 77y., but Xm is not w-covered by the family 17k, i.e. Xm\X(f < n) 0, whence f (x) > n > m., for some x E Xm. The definition of Xm now gives

fOC.

But then C fl A = U{C fl A,,: k < m} is closed in C, a contradiction. We now finish the proof of theorem 3.7.

11.3.9. Lemma. If Cr,(X) is n, k.-space, then X I- e. Proof. It is well known that ind X = 0 if Cp(X) is a k-space. Assume that C. is not fulfilled in X. Take a corresponding open w-cover i. Put A = (f E C,,(X): f (X) C D and Z(f) = f -1(0) can be w-covered by a countable subfamily of 77}. It is readily seen that f E A\ A, where f - 0, i.e. A is not closed in Cp(X ). However, the closure of any countable subset of A lies in A. Indeed, let B C A, IBS = 1 o, and g E B. There is a countable family rIO which w-covers Z(f) for any f E B. If F is finite and F C Z(g), then, since g E B, there is an f E B such that F C Z(f). Hence rlo is an w-cover of Z(g) too, i.e. g E A. We now choose a compact subspace C C Cc(X) such that c fl A is not closed in C. Let g E C fl A \ A and S = Z(g). It is obvious that g maps X into D, and hence S is an open-closed subset of X. while S is not w-covered by any countable subfamily of r).

Note that C fl A is countably compact, for if H C C fl A and H is countable, then the closure of H in the compact, set C is contained in C fl A. We construct by transfinite recursion a sequence {(fe, 71F, F4+1): < wl} with the following properties: a) fF E C n A, r)F C 71 is countable, FF}1 is finite, and Fe+1 C S; b) 'F, c 'g%7 if C1 < e2 < wl; c) ?IF is an w-cover of Z(ft), but not of F4+1; d) If S1 + 1 < %2 < wl, then FF,+1 C Z(f{s).

Let fo E C fl A be arbitrary,'io C q a countable w-cover of Z(fo). Since 710 is not an w-cover of S, there is a finite FI C S such that no element of 'nl contains FI. Let a < w1, and let (fe, rIF, FF+1) be constructed for each i; < a. We consider two cases. Case 1) a is a limit ordinal. Choose Cn -> a and take f,, to be a limit point of the set { ff,,: n E N+} in C fl A, Ice = U{r1F: < a), and F0,+1 a finite subset of S not w-covered by the family r/a.

Case 2) a = ,6 + 1. Note that if M C S is countable, then there is an f E C n A

56

11, DUALITY BETWEEN INVARIANTS OF LINDELOF NUMBER AND TIGHTNESS TYPE

that vanishes on M. In fact, if M = {x,,: n E N+} C Z(g), then, since g E G n A, for each n e N+ there is an f E C n A that vanishes at the points {xk: k < n}. Now we take f to be an arbitrary limit point of the set { f,,: n E N+} in C n A. Take a function fa E C n A that vanishes on the set U{Ff+1: !5,8}, a countable family rla C 77 such that 77. is an w-cover of Z(f,,) and 77p C rja, and, finally, a finite set F.+1 that is not w-covered by 77,,,.

To finish the construction, we find a complete accumulation point h E C for the sequence {ft: f < w1}, and put T = Z(h) n S. For shortness reasons we write Zf instead of Z(ff). Note that Fa+1 C Zp if and only if a + 1 < /j. Indeed, if F.+1 C Z, , then 770 is an w-cover of Fa+i, hence a + 1 < /3. On the other hand, if a + 1 < /I and a is an isolated ordinal, then fp vanishes on Fa+1, i.e. Fa+1 C Zp. If ,6 is a limit ordinal

and a + 1 < Q, then, since C. - 0, there is an no E N+ such that a + 1 < Cn and F.+1 C Zf, for all n > n0; hence Fa+i C Zp. Since h is a complete accumulation point of {ft: f < w1}, we have U{F4+1: r; C W1} C T.

We show that for {Ff+1: e c w1} the following condition holds:

3) let K C X be finite. Then there is an open set G D K in X such that { < wi : F(+1 C G}) < l io.

Indeed, since T is open-closed and U{Ff+i: C C w1} C T, it suffices to prove 3) for K C T. For each finite K C T there is a minimal aK < wi such that K C Z,,,,,. It is easily seen that aK cannot be a limit ordinal. Put Ifo = K, Co = ,OKo, where aKo = ,OK0 + 1. It is possible that aKo = 0, i.e. 6K,, is not defined. However, this situation is trivial in view of the fact that we may then set G = Zo. Let K1 = ZfonKo, CI = 6K1, if 79K1 is defined, i.e. if aK1 = 13K, + 1, etc. Our construction finishes on one

of the following two grounds: either a chain Ko D Ki If = 0 is obtained, or K. C Zo for a certain n E N+. We now put Ck = (Zf,,+1 \ ZZ,k) n n{Zef+1: j < k} and G = U{Gk: k < n}. Clearly, G is open (recall that i;k + 1 is the smallest of the ordinals C such that Kk C Zf). If x E K and the first case holds, then.x E Ki \ K1+1, where i < n. Then x E K; for

j
XE(Ze;+i\ZZ;)nn{Zc,+i: j
Put 77 = {G C X: G is open and l{C < wi: Ft C G}1 < n}, 77 = U{77,,: n E N+}. Using property 3) we thus obtain that 77 is an open w-cover of X, and ljn C for

all n E N.

4. HEWITT-NACHBIN SPACES AND FUNCTIONAL. TIGHTNESS

`.7

Thus, there is a sequence X -+ X such that By lemma 3.8 we have X But this is impossible: if X is w-covered by the X is w-covered by the family family 77,,, then X contains at most n sets FF+I. However X -- X implies that some X contains BI sets Ff+I. This contradiction finishes the proof of lemma 3.9 and theorem 3.7. Not solved is the following

11.3.10. Problem. Let Cp(X) and CP(Y) be Frechet-Urysohn spaces. Is it then true that CC(X) x C9(Y) is a Th chet-Urysohn space? As noted in chapt. II, §1, the corresponding answer for the tightness of function spaces is in the negative.

4. Hewitt-Nachbin spaces and functional tightness A special and, as it turned out, important. form of topological completeness was introduced by E. Hewitt and L. Nachbin. Spaces satisfying this condition of completeness are called functionally closed spaces, Q-spaces, realcompact spaces, R-complete spaces, or Hewitt-Nachbin spaces (see [16], [66], [105]). Recall that a space is called R-complete (or realcomplete) if it is homeomorphic to a closed subspace of the space

R' for a certain r. If there does not exist an Ulam measurable cardinal (see chapt. 0), then a space is Rcomplete if and only if it is Dieudonne complete [66]. Here, a space is called Dieudonne complete if it is complete in the largest uniform structure corresponding to its topology. The Dieudonne complete spaces can be characterized as the homeomorphic images of closed subspaces of a product of metric spaces (see [66]).

The following characterization of R-completeness in the language of continuous functions is useful.

11.4.1. Proposition [16], [66]. A space X is R-complete if and only if for any space Y D X and point f E Y \ X lying in the closure of X there is a function f E Cp(X) that cannot be extended to a realvalued continuous function on the space

XU{f}. Clearly, the product of an arbitrary set of R-complete spaces is R-complete. In particular, all Lindelof spaces are R-complete. In view of M. 0. Asanov's theorem (see chapt. 1, §4) it may be conjectured that the tightness of a space X is countable if and only if Cp(X) is R-complete. Litterally taken, this conjecture is not true, but

a nontrivial version of it turns out to be true. In particular, it turns out that if t(X) < No, then Cp(X) is R-complete. Also in relation to the following fact it is useful to have R-completeness criteria for spaces: every R-complete pseudocompact space is compact [66]. This can prove useful in the study of compactness properties of sets of functions in CP(X) and in the construction of such sets.

58

II. DUALITY BETWEEN INVARIANTS OF LINDELOF NUMBER AND TIGHTNESS TYPE

Note that a space C,,(X) is R-complete if and only if it is Dieudonne complete. This follows from the fact that the Suslin number of CC(X) is countable [16]. A map f : X ---* Y is called r-continuous (where r is a fixed cardinal) if for every

subspace A of cardinality at most r the restriction AA: A

Y of f onto A is

continuous.

The functional tightness te(X) of a space X is the smallest (infinite) cardinal r such that every realvalued r-continuous function on X is continuous [75]. The following technical concept turns out to be useful in the study of functional tightness [75]: the weak tightness t,(X) of a space X is the smallest (infinite) cardinal r such that the following condition is fulfilled:

if a set A C X is not closed in X, then there are a point x E A \ A, a set BC A, and a set C C X for which: XEB,BCC,and (CI
11.4.2. Proposition. For every space X: a) te(X) tc(X);

b) 4(X):5 IM;

c) t,(X) < d(X).

Proof. Inequalities b) and c) are obvious. We prove a). Put r = t,(X), and consider an arbitrary realvalued r-continuous function f on X.

Let P C X. We must show that f (P) C T - Put A = {x E P: f (x) E f (P)}. Let Clearly, P C A C A = P. If A = P, our aim is reached. Let A # P. Then A # A, and, since t,(X) :5,r, there are x E A \ A, B C A, and C C X as in the definition of weak tightness: x E B, B C C, and ICI _< r.

Take an arbitrary open set U in R for which f (B) C U, and put Co = {c E C: f (c) E U} and CI = C \ Co. Let b E CI. Since f is r-continuous and [CI I _< r, we have f (b) E f (CI). But f (CI) f1 U = 0 and U is open in R. Hence, f (b) V U. Now f (B) C U implies that b 0 B. So, CI n B = 0. Since BCC = Co U CI, we have B C Co and x E B C Co. By the r-continuity of f, the conditions x E Co and ICo) < r imply that f (x) E f (Co). But f (Co) C U. Thuf) E U. The space R is regular, hence fl{U: U open in R and U D f (B)} = f (B). We obtain that f (x) E L (B), and, hence, f (xZE_f (A) (since f (B) C f (A)). By the definition of A, f (A) C f (P). Hence f (x) E f (P). But x E A = P. Using the definition of A once more, we conclude: x E A, contradicting x E A\ A. Thus A = P and f is continuous,

i.e. ta(X) < r. 11.4.3. Corollary. Always, to(X) < d(X). 11.4.4. Example. Weak tightness (let alone functional tightness) does not coincide with tightness. In fact, the tightness of the space R`, where c = 2"O, is equal to c. At the same time, R` is separable [16], hence te(R`) = t,(R`) = l o.

11.4.5. Example.

Consider the space T(WI) = {a: a < wI} of all ordinals not

4. HEWITT-NACIIBIN SPACES AND FUNCTIONAL TIGHTNESS

59

exceeding w1, in the ordinary (i.e. order) topology. Put f (a) = 0 for all a < wl and R thus defined is clearly No-continuous, but not f (w1) = 1. The function f : T((,,t1) continuous. Hence, the functional tightness of T(w1) is KI. Moreover, t,(T(w1)) = RI.

The space T(w1) is homeomorphic to a closed subspace of the separable space Rt. We have te(T(w1)) = 3II > No = te(R`), and t,(T(w1)) = RI > No = t,(R`). Hence neither functional tightness nor weak tightness are monotone, generally speaking, with respect to closed subspaces, in distinction to tightness, which is monotone with respect to arbitrary subspaces. It would appear that the notions of T-continuity and functional tightness may claim to be the most adequate expression of certain ideas. However, they allow a curious modification which, as subsequent results will show, is also an important concept. A map f : X - Y is called strictly T-continuous if for each set A C X with 1AI < r there is a continuous map g: X -p Y for which 91A = f IA (i.e. f (x) = g(x) for all x E A). The weak functional tightness (or R-tightness) tR(X) of a space X is the snlalle$l, infinite cardinal r sueli that evet;y realvalued strictly r-cont.inttt>us fnrt.iun on X is continuous [751, 1111.

It is obvious that every strictly 7-continuous function is -r-continuous. Hence we have

11.4.6. Proposition. Always, tR(X) < te(X). In relation to the nonmonotonicity of the functional tightness (see example 4.5), the following theorem is interesting.

11.4.7. Theorem. For any space X the following conditions are equivalent: a) tR(Y) < r for all Y C X; b) te(Y) < r for all Y C X; c) t (X) < T.

Proof. Clearly, c) implies b) and b) implies a). We derive c) from a).

Consider an arbitrary A C X. We show that the set P = U{R: B C A and )B) :5,r) is closed in X. Assume the contrary, take ay E A \ P, and put Y = P U { y}. We show that the function f : Y --. R defined by f (x) = 0 for all x E P and f (y) = 1, is strictly r-continuous. Let C C Y and Cl 1< r. Put Co = C fl P. It easily follows

from 1Co1 < T and Co C P that y 0 Co. Hence there is a realvalued continuous function g on Y such that g(y) _= 1 and g(x) = 0 for all x E Co.

Clearly, f Ic = g1 c, i.e. f is strictly r-continuous. But tR(Y) < r, hence f is continuous. This contradicts the fact that y E P, f (y) = I, and f (P) = {0}. Quite recently E. A. Reznichenko has constructed a space whose weak functional tightness is not equal to its functional tightness. In relation to this we note

11.4.8. Proposition. If a space X is normal and r is an arbitrary cardinal, then a) every realvalued T-continuous function f on X is strictly r-continuous;

60

11. DUALITY BETWEEN INVARIANTS OF LINDELOF NUMBER AND TIGHTNESS TYPE

b) tR(X) = te(X). Proof. Clearly, a) implies b). We prove a). Let A C X and Al C< -r. By corollary 4.3, te(A) < d(A) < JAI < r. The function f 1a is r-continuous on 7, hence continuous. Since X is normal, there is a realvalued continuous function g: X -+ R such that gl-x = In+ and so 91A = PA. Thus f is strictly r-continuous. This reasoning shows that if tR(X) = r and every closed subspace F with d(F) < r is C-embedded in X, then t8(X) = r = tR,(X). We say that A C X is a set of type G, in X if there is a family -y of open sets in X such that A = fly and Jyl < r. A set A C X is called r-placed in X if for each point

xEX\Athere isasetPoftype G,inXsuch that xEPCX\A.

11.4.9. Proposition. Let X C Y C Z, with X -r-placed in Y and Y r-placed in Z. Then X is r-placed in Z. The proof of this assertion is obvious.

11.4.10. Proposition. If a space X is r-placed in some compactification bX of it, then X is 7--placed in

X.

For the proof it suffices to recall that 6X can be continuously mapped onto bX in such a manner that f -1(bX \ X) = /3X \ X. Put q(X) = min{r > l o: X is r-placed in )3X}; q(X) is called the Hewitt-Nachbin number of X. We say that X is a Q,-space if q(X) < r. It is well known that q(X) < 1 o if and only if X is realcomplete; this explains the terminology.

11.4. 11. Proposition. If X C Y and X is closed in Y, then q(X) < q(Y). This readily follows from 4.9. Recall that a canonical closed set is the closure of an open set. A space X is called an m,-space, where r is a given cardinal, if for each canonical

closed set F in X and each point x E F there is a set P of type G, in X such that

XEPCF.

Clearly, X is an m,-space for r = IXI. This allows us to give the following definition:

put m(X) = min{r > l o: X is an m,-space}. The space X is called a Moscow space if m(X) < 1 o. If every canonical closed set in X is of type G then X is a Moscow space. Hence the space RA, as well as all everywhere dense subspaces of it, is a Moscow space. Therefore we have

11.4.12. Proposition. Always, m(C,(X)) <;Zo and m(LP(X)) < 1Zo. The following result is, in essence, due to A. Ch. Chigogidze.

4. HEWITT-NAC'HBIN SPACES AND FUNCTIONAL TIGHTNESS

11.4.13. Proposition.

61

Let X C Y, q(X) < r, X = Y, and m(Y) < r. Then X is

r-placed in Y.

Proof. Let B be an arbitrary compactum for which Y C B. There is a continuous map f: OX -->Bsuchthat f (x) = x for all x E X (16]. LetyEY\X. Then X = Y and the closedness of the compactum f (QX) in the Flansdorif space B imply that f(/3X) D Y. Hence If''(y)I > 1. Case I. Let I f-'(y)I = 1, i.e. f-'(y) = (z) for some z E /3X \ X, since f-'(X) = X and y 0 X. The condition q(X) < r allows us to choose a family ry of open sets in /3X such that fly C )3X \ X and z E n-y, IryI < r. Then f -I(V) C ny, hence y na, where A = [B\ f (/3X \U): U E y}, JAI < IryI < r, and every V E A is open in B.

We have nA=B\U{f(,OX\U): UEy}=B\f(U{)3X\U: UEy})CB\f(X)= B \ X, since f (X) =X and (ny) n X = 0. Put P= (nA) n Y. Then y E PC 1'" \ X and P is a set of type C, in Y. Casr, 11. If-'(y)1 >_ 2. Fix Z1, z2 E f-'(y), z, / z2. Take neighborhoods Oz, and Oz2 of z, and z2 in OX such that Oz, n Oz2 = 0, and

put l;=OznXand F;=clyVi, i=1,2.

Clearly, zi E clax U. Since the map f is continuous, f (V j) = V, and y = f (zi), i = 1, 2, we have y E F, n F2. Moreover F1 n F2 n X C cl0x (Oz,) n cl0x (Oz2) = 0.

Thus, FinF2CY\X. Take now open sets Wi in Y for which W; n X = Vi, i = 1, 2. Since X = Y, we have Fi = cly(Wi), i.e. the F; are canonical closed sets in Y. Now in.(Y) < r implies that there are families ryi of open sets in Y such that y E ny; C F; and Iryi1 < r, i = 1,2.

For-y =yjU-y2we then have l-yI
Proof. Let g E RY \ Cp(Y). Then tR(Y) < r implies that there is a set A C Y such that IAI < r and 9IA f IA for all f E CC(Y). Consider the restriction map 1r: RY _ RA, i.e. lr(h) = hIA for all h E RY. The set Z = ir(C9(Y)) is r-placed in RA, since IAI < r (all singleton subsets of RA are of type G,). It follows from ir(g) = 91A Z that there is a set Po of type G, in RA for which 7r(g) E Po C RA \ Z. Then P = it-1(Po) is a set of type G, in RY and g E P C R' \ Cp(Y).

I1.4.15. Proposition. If Cp(_K) is r-placed in RX, then tR(X) < r. Proof. Let g be a strictly r-continuous function on X, and P an arbitrary set of type G, in RI such that g E P. There is clearly a set A C X for which IAI < r and g E If E R: f IA = 9IA} C P. Since g is strictly r-continuous, there is an h E C,,(X) such that hIA = 9IA. Then h E P. So, P n CC(X) # 0 for any set P of type G, in Rx containing g. Since CC(X) is r-placed in RX, we conclude: g E Cp(X), i.e. g is continuous. Hence tR(X) < r.

62

11. DUALITY BETWEEN INVARIANTS OF LINDELOF NUMBER AND TIGHTNESS TYPE

II.4.16. Theorem (A. V. Arkhangel'skiT [751). For any X, tR(X) = q(Cp(X)). Proof. Put r = tR(X). By proposition 4.14, Cp(X) is -r-placed in Rx. But Rx is realcomplete, hence [16] No-placed in f3RX. Thus (proposition 4.9), Cp(X) is r-placed in ,QRX. This implies by proposition 4.10 that Cp(X) is r-placed in f3Cp(X). Thus,

q(C9(X)) S r = tR(X) Put A = q(CC(X)). We have m(RX) = Po < A and CC(X) = Rx. Proposition 4.13 implies that C9(X) is A-placed in Rx. Applying proposition 4.16 we conclude that tR(X) < A. So, tR(X) < q(CC(X)), and hence tR(X) = q(CC(X)).

11.4.17. Corollary. The weak functional tightness of a space X is countable if and only if Cp(X) is realcomplete.

II.4.18. Corollary [75].

If the spaces Cp(X) and Cp(Y) are homeomorphic, then

tR(X) = tR(Y).

11.4.19. Corollary. If X is normal, then te(X) = q(Cp(X )). This follows from theorems 4.8 and 4.16.

II.4.20. Corollary.

If the spaces X and Y are normal and Cp(X) and Cp(Y) are homeomorphic, then te(X) = to(y). In relation to assertions 4.18 and 4.20, the following is of interest:

11.4.21. Corollary. Always, q(X) < tR(Cp(X)). Proof. By theorem 4.16 we have tR(Cp(X)) = q(CPCp(X)). But X is homeomorphic to a closed subspace of CpCp(X) (see chapt. 0). Thus (proposition 4.11), q(X) < q(CpCp(X)) < tR(C,(X)).

11.4.22. Theorem (V. V. Uspenski [153]). Always, te(CC(X)) < q(X). Below we use the following notations: if A C X and r is a cardinal, then [A]T = U{B: B C A and IBS < r}. To prove the theorem we need the following

11.4.23. Lemma.

Let f : Y -. Z be a continuous map, f (Y) = Z, and te(Y) < r. Let, moreover, there be a base B of Y such that for each set V E B there is an open set G in Z satisfying f (V) C G C If (V)I,. Then te(Z) < r.

Proof of the lemma. Let 0: Z --b R be a r-continuous function. Take arbitrary zo E Z, yo E f-I(zo), and e > 0. The map 45o f: Y -> R is r-continuous, hence

4. HEWITT-NACHDIN SPACES AND FUNCTIONAL TICHTNESS

63

continuous since te(Y) < r. Therefore there is a V E B such that yo E V and cb(f (V)) C [4(zo) - e, 4(zo) + el. Since 0 is r-continuous, we have ¢((f (V)],) C 10(zo) - e,Qi(zo) + E].

By requirement, zo E f(V) C U C [f (V)], for some open set U in Z. We have 0(U) C O(lf(V)],) C [0(zo) - e,¢(zo) + e]. Hence ¢ E Cp(Z). The lemma has been proved.

Proof of theorem 4.22. Put C = Cp(X, (0,1)). Clearly, Cp(X) is homeomorphic

to C. We show that te(C) < r, where r = q(X). Consider the space Y = {g E Cp(/3X, [0,1]): -Q-'(0) U 9-'(1) C pX \ X). The restriction map 7r: Y -+ C (where ir(g) = glx for all g E Y) is continuous from Y onto C. We have t9(Y) <- t(Y) < t(CC(,QX)) = lio. By lemma 4.23 it suffices to indicate a base B of Y such that the set V I x = If IX: f E VI is contained in the interior of the set (V l x]T for all V E S. It turns out that this property holds for the standard base of Y consisting of all sets of the form

V={qeY:

j=l,...,m.},

where x;EXfori.=1,...,kandxxE/3X\Xforj=l,...,m,while nonempty open subsets of the interval [0, 11.

We establish that [V lxl, D If E C: f (xi) E Oi, i = 1, ... , k}. Then the proof is finished, since at the right side stands an open set in C containing VIx.

So, let f E C and f (xi) E O; for i = 1,...,k. Put K = {xi, ... , x;,,}. Since q(X) < r and K is a finite set lying in OX \ X, there is a family ry of open sets in 13X with the properties: a) Iyl <,r;

b) Kcny; c) for each finite set T C X there is a U such that U nT = 0. We assume that -y = {Ua: a E M}, where IMI < r, and put P,,,,n = {x E X \ Ua: f (x) E [1/n,1-(1/n)]}. Since I is continuous and f (X) C (0,1), the set Pa,,, is closed in X, and U{Pa,,,: a E M, n E N+} = X (see condition c)). Clearly, K n where Pan is the closure of Pa,,, in /3X. Take a function f E C,(/3X) for which fix = f . Fix also r,, E O' n (0, 1), for

j = 1, ... , m. Choose n' E N+ such that 1/n' < r; < 1 - (1/n') and 1/n' < f (x,) < 1 - (1/n*) for all j = 1,...,m, i = 1,...,k. Clearly !(-P...) C [1/n, l - (1/n)]. Since K is finite and K n Pa,, = 0 (by b)), while /3X is normal, there is for each a E M and n > n', n E N+, a continuous function fa,,,: /3X -+ [1/n, 1 - (1/n.)] such that fa,,,(x;) = rr, for j = 1,...,m, and fa,nlP,,, = flee,,,. By condition c) we may assume that {x1,...,xk} C Pa,,, for all

aEMandn>n`,nEN+. Clearly, the set B={fa,,,: a E Mandn>n',nEN+} is contained in V. We have IBI < IMI < r. Finally, condition c) together with the definition of fe,,, imply that f t=- B. So, f E B C [Vl,r]T, as required. The theorem has been proved.

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11. DUALITY BETWEEN INVARIANTS OF LINDELOF NUMBER AND TIGHTNESS TYPE

The inequalities in 4.21 and 4.22 imply

te(CP(X)) :5 q(X) < tR(Cp(X)) :5 te(CP(X)) Thus we have the final

11.4.24. Corollary [153], [11], [75]. Always, q(X) = tR(C'(X)) = te(Cp(X)).

II.4.25. Corollary [153], [11], [75].

If the spaces CP(X) and Cp(Y) are homeo-

morphic (i.e. X L Y), then q(X) = q(Y). In particular, if X is realcomplete, then Y is also realcomplete.

11.4.26. Theorem (V. V. Uspenskii [153]).

Assume that there does not exist an (Ulam) measurable cardinal. Then te(CP(X)) < No for all metric spaces X; in particular, te(RX) = po for all X. This result shows how different functional tightness and tightness can be. I lowever, there is the following useful result.

11.4.27. Proposition [53]. If tR(X) < r and a point x E X is not isolated in X, then there is a set A C X \ {x} such that IAA < r and x E A.

Proof. Assume the contrary. Then the function f : X - R defined by f (x) = 1, f (y) = 0 for all y E X \ {x} is strictly r-continuous, as can be readily verified. Thus, tR(X)) > r, contradicting the assumptions. Let X be an arbitrary space, f3X its Stone-Cech compactification, and r a cardinal. By vTX we denote the subspace of OX consisting of all x E ,QX for which every set of type GT in QX containing x intersects X. It is easy to prove [141

II.4.28. Proposition. For any space X the following assertions are true: a) q(v,X) < r; and b) every continuous function f : X -+ R can be extended to a realvalued continuous function on all of vTX.

11.4.29. Theorem.

For any space X the topology of the space CP(v,X) is the strongest topology on C(X) that induces on every set B C C(X) with JBI < r the

same topology as that induced by the topology of CP(X).

Proof. The above implies that the realvalued continuous functions on vTX can be canonically identified (using proposition 4.28) with such functions on X. Now 4.28a) and 4.22 imply that te(Cp(vTX)) < r. We then need

11.4.30. Lemma.

If te(Y, T) < r and T' is a topology on Y not contained in T, then there is a set B C Y for which (BJ< r and TSB # TIB.

Proof of the lemma. Since T

T, there is a realvalued continuous function g on

4. HEWITT-NACHnIN SPACES AND FUNCTIONAL TIGHTNESS

65

(Y, T') which is discontinuous on (Y, T). Now te(Y,T) < r implies that there is a set B C Y such that IBS < r and 91B is a discontinuous function on (B,TlB). Since gIB is continuous on (B, V I B ), we clearly have TSB # T I B. The lemma has been proved. By lemma 4.30, the topology of Cp(v,.X) contains every topology on C(X) that coincides on all sets of cardinality < r with the topology generated by CC(X). Therefore, the proof of theorem 4.29 will be finished if we prove that for a B C C(X) with [B[ < r the topology TB generated on B by Cp(vrX) coincides with the topology TB generated on B by Cc(X). Since X C vrX, we have TB C TB. We prove the opposite inclusion. So, it remains to prove that if h E C(X), B C C(X), CBI < r, and h V IT (in Cp(vrX)), then h also does not belong to the closure of B in the space CC(X). Let

V(xl,...,x,n,h,e) = {f E C(X): I f(x,) - h(xr)) < e, i = 1,...,m} be a

standard neighborhood of h in (,(vX) not intersecting B.

The set th = h-Ih(xi) is of type G6 in vX, since h: vrX -+ R is continuous. Similarly, the set 4ii = f -If (xi) is of type G6 in vrX for every f E B. Hence the set Si = n{V nV: f E 131 is of type Cr in vrX (we use the fact that 1131 < r). Clearly, ;ri E .51, i = 1, ... ; 9n..

The definition of the space v X now implies that there are xi E X n S; for i. _ 1, ... , in. Clearly, h(xi) = h(d1) and f (xi) = f (.ti) for all i = 1, ... , in and all f E B. This implies that im, h, e) n B = 0. Thus, h (in Cp(X) ). The theorem has been proved. Since in theorem 4.22 the topology of Cp(vrX) is described in terms of the topology of CC(X), we obtain

11.4.31. Corollary (0. G. Okunev). If the spaces Cp(X) and CC(Y) are homeomorphic, then the spaces CC(v X) and CC(vY) are homeomorphic. We can similarly prove

11.4.32. Corollary.

If the spaces Cp(X) and Cp(Y) are linearly homeomorphic, then the spaces Cp(vrX) and Cp(vY) are linearly homeomorphic.

In the study of duality between the Hewitt-Nachbin number and the weak functional tightness, the following two simple assertions can be successfully used.

II.4.33. Proposition. te(Y) < r if and only if the space Y can be represented as the image under an R-quotient map (see chapt. 0) of a space whose local cardinality does not exceed r. 11.4.34. Proposition. te(Y) < te(X ).

If f : X -+ Y is an R-quotient map, f (X) = Y, then

The proofs of these assertions are easy, and follow a well-known general scheme (see, e.g., [66], [16]).

In particular, 4.33 obviously implies that if te(X) < No, then Cp(X) can be embed-

. DUALITY BETWEEN INVARIANTS OF LINDELOF NUMBER AND TIGHTNESS TYPE

66

ded as a closed subspace in a product of spaces with a countable base (and is hence R-complete). In conclusion to this section we note some consequences of the results obtained above.

11.4.35. Corollary. Always, q(X) = q(C,,C,,(X)) = q(Lp(X)). In particular, every R-complete space can be embedded as a closed subspace in an R-complete linear topological space.

11.4.36. Corollary. Always, tR(X) = tR(CpCp(X)) = te(CCCC(X)). In particular, every space with countable functional tightness can be embedded as a closed subspace in a linear topological space of countable functional tightness. These consequences are especially interesting in relation with the fact that there is yet no answer to the following questions: 11.4.37. Problem. Which spaces of countable tightness can be embedded (as closed subspaces or in another way) in a linear topological space of countable tightness?

11.4.38. Problem. Which Lindelof spaces can be embedded as closed subspaces in a Lindelof linear topological space? Assertions 4.16 and 4.24 imply

11.4.39. Corollary.

If Xt' Y for some n E N}, then q(X) = q(Y) and tR(X) _

tR(Y).

5. Hereditary separability, spread, and hereditary Lindelof number It is well known that, assuming the continuum hypothesis, we can construct a nonseparable space every subspace of which is Lindelof. Assuming the continuum hypothesis it is also possible to obtain a non-Lindelof hereditarily separable space. On the other hand, the following assertion is compatible with the Zermelo-Fraenkel axiom system ZFC: every hereditarily separable space is hereditarily Lindelof. A new view on the relation between hereditary Lindelofness and separability is given by the functional approach. Recall that the spread of a space is the supremum of the cardinalities of its discrete (in itself) subspaces. The spread of X is denoted by s(X). Clearly, both hereditarily separable and hereditarily Lindelof spaces have countable spread. We describe a general construction, related with the transition from sets in CC(X) to sets in X.

11.5.1. General construction. Let l3 be a countable base of R, A C CC(X), and let for each f E A a neighborhood G1 of f in CP(X) be fixed.

5. HEREDITARY SEPARABILITY, SPREAD, AND HEREDITARY LINDELOF NUMBER

67

For f E A we choose n(f) E N+, xf = (xi , ... , xn(f)) E X"(f), and

Vf = (Vf,..., n(f)) E B°(/) = B

8, n(f) times

such that f E Wf = {g E C7,(X): g(x[) E Vf, i = 1,...,n.(f)} C Cf. The set U(f) = {(xl,...,xn(f)) E Xn(f): f(xi) E Vf, i = 1,...,n(f)} = f-'(Vf) X x f-'(V (f)) is open in X"(-'), and xf E U(f).

FornEN+andV=(V1i...,Vn)EB"we putA(n,V)={f C- A: n(f)=nand Vf = V}. Clearly, U{A(n,V): n E N+ and V E B"} = A. Using this notation, the following simple assertions hold.

11.5.2. Lemma. I f f, g E A(n, V) and x9 E U(f), then f E C C.

Proof. U(f) 3x9 = (.r. , ... , x) implies that f (,T,) E

i. _ ], ... , n. l fence, by the

dcfinit.ioll of W,0, f E W. C CO.

11.5.3. Lemma. Let f, g E A(n, V) and either f ¢ G9 or g 0 C f. Then xf # x9. Proof. Suppose x9 = xf. Then U(f) 3 xf = x9, and f E G9 by lemma 5.2. Similarly, U(g) D x9 = xf, and hence g E Gf. Define the map 0 _ 0(n,v): A(n,V)

Y(n,V) = {xf : f E A(n,V)}

by 4(f) = xf, f E A(n, V). Lemma 5.3 implies

II.5.4. Proposition.

Let n E N+ and V E 13n. If for distinct f,g E A either f 0 G. or g 0 Gf, then q5: A(n, V) -> Y(n, V) is bijective. We now consider a concrete Situation.

11.5.5. Proposition. If s(X°) < T for all n E N+, then also s(C,(X)) < T. Proof. Let A be an arbitrary discrete (in itself) subspace of CC(X). For each f E A we take Of to be an arbitrary neighborhood off in CP(X) such that Cf n A = (f }. We use the notation from the previous construction. Let n E N+ and V E Bn. By lemma 5.2, if f, g E A(n, V) and x9 E U(f), then f E G9. But G9 fl A = {g}.

and f E A. Thus f E {g}, i.e. f = g and x9 = xf. So, IA(n,V)l _ JY(n,V)J and U(f) fl Y(n, V) = {-TI } for all f E A(n, V). Since U(f) is open in Xn, we conclude that the subspace Y(n, V) is discrete. Hence, IA(n,V)f = IY(n,V)J < s(Xn) < r;

moreover, n E N+ and V E 8' are arbitrary. But T > Ho and A = U{A(n,V): n E N+, V E MI. Hence, since IB"1 < No, (AI < r No = T.

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It. DUALITY BETWEEN INVARIANTS OF LINDELOF NUMBER AND TICHTNESS TYPE

11.5.6. Proposition. If s(X") < r for all n E N+ then also s((CC(X))") < r for all n E N+. Proof. For n E N+ we put Y = X x {1, ... , n}. The spaces (CC(X))" and C9(Yn) are clearly homeomorphic. It is obvious that s(Yn) = s(Xk) < -r for k E N+ (as agreed upon, the values of cardinal invariants are always infinite). Applying proposition 5.5 we obtain n E N+. S((Cp(X))") = s(Cp(Yn)) C r,

II.5.7. Theorem.

Always, r = A, where r = sup{s(X"): n E N+} and A _

sup{s((CC(X))"): n E N+}.

Proof. By proposition 5.6, \ < r. Put CC(X) = Y. Applying proposition 5.6 again we obtain s((Cp(Y))") < for all n E N+. But X C CCP(X) = Cc(Y), and hence X" C (Cc(Y))". We conclude: s(X") < s((CI,(Y))") < A for all n E Nt. Hence

r=sup{s(Xn): nENi}
We now turn to the comparison of hereditary separahility and hereditary Lindclofness.

In the sequel, as usual, hl(X) = sup{l(Y): Y C X} and hd(X) = sup{d(Y):

YCX}.

Recall that a space X is called left (right) if X can be well ordered in such a way that every left (respectively, every right) ray is closed in X. Such a well order on X is called left (right). (A left (right) ray in an ordered set (X, <) is a set A C X such

that y<xandxEA(respectively,x
II.5.8. Proposition.

If FYI < r for every left (right) subspace Y of the space X" and for all n E N+, then IAA < r for every right (respectively, left) subspace A of

CC(X).

Proof. Consider A f = {g E A: g < f) (Af = {g E A: f < g}), where < is the left (right) well order on A. Since A f is a left (right) ray, it is closed in A. Let Gf = Cp(X)\Af. Clearly, Gf is open in Cp(X) and contains f. For each n E N+ and V E B" we define A(n, V) C A (corresponding to the sets Gf chosen by us). The map

0: A(n,V)

Y(n,V) is bijective, since for f' < f" we have f 0 Gf. (f" 0 Gf-). Hence we can introduce on Y(n, V) a well order by: xf < xQ if and only if f < g.

5. HEREDITARY SEPARABILITY, SPREAD, AND HEREDITARY LINDEL6P NUMBER

69

Consider an arbitrary left (right.) ray Z C Y(n, V). Let xJ E Z. Then U(f) n Y(n,V) C Z. In fact, if x9 E U(f) for some g E A(n, V), then f E G9, hence g < f (respectively, f < g). Hence every left (right) ray is open in Y(n, V), i.e. Y(n, V) is a right. (left) subspace of X". Therefore (A(n,V)J = IY(n,V)J < r for all n E N+ and V E B", giving JAI

=r.

11.5.9. Proposition.

If IYJ < r for every left (right) subspace Y of the space X" and for all n E N+, then JAI < r for every right (respectively, left) subspace A of the space (CC(X))", for all n E N+. Proof. Let Y" = X x { 1, ... , n} for all n E N+. Then, clearly, (Cc(X ))" is homeomorphic to C,(Y"). However, for every natural number n the cardinality of an arbitrary left (right) subspace Y,k does not exceed r (because Yk is homeomorphic to the direct sung of nk copies of X'). It remains to apply proposition 5.8.

11.5.10. Theorem (Ph. Zenor [159], N. V. Velichko). a) sup{hd(X"): n E N+} = sup{hl((CC(X))"): n E N4 }; b) sup{hl(X"): n E N+} = sup{hd((CC(X))"): n E N+}.

Proof. Put r1 = sup{hd(X"): n E N+}, r2 = sup{hl(X"): n E N+}, A, = sup{hl((Cp(X))"): n E N+}, A2 == sup{lid ((Cc(X))"): n E N+}. Using the equalities

hd(Z) = sup{IYJ: Y C Z and Y a left space} and hl(Z) = sup{ FYI: Y C Z and Y a right space}, and applying proposition 5.9, we find that Al < r1 and A2 < r2. Put. Y = Cp(X). Applying proposition 5.9 once more, we obtain that hl((C,,(Y))") < A2 and hd((CG(Y))") < Al for all n E N+. But X is homeomorphic to a subspace of Cp(Y), hence X" is homeomorphic to a subspace of (Cp(Y))". Thus, hl(X") < A2 and hd(X") _< Al (for all n E N' ). Therefore r2 < A2 and r1 < A1i which proves the inequalities a) and b). It is not true, in general, that :,(X) = s(Cp(X)), hd(X) = hl(Cp(X)), and hl(X) hd(CC(X)).

II.5.11. Example.

Let X be the `two arrows' compactum. It is hereditarily

separable and hereditarily Lindel6f. However, Cp(X) does contain a discrete subspace of cardinality 210, even a closed one (see example 1.7).

In relation with example 5.11 the partial strengthenings of theorems 5.7 and 5.10, given below, are of some interest.

11.5.12. Second fundamental construction.

Suppose we are given a space X, and n E N+, and a subspace T" = T"(X) with the following properties: a)

(with OP1U...UP");

b) every Pi is open in T" (and not empty);

c) Pi n Pj = 0 if i#j;

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11. DUALITY BETWEEN INVARIANTS OF LINDELOF NUMBER AND TIGHTNESS TYPE

d) for each point x E X, a neighborhood Ox of it in X, and all i = 1, ... , n, there is a continuous map f : X T. such that x E f -'P, c f -' Pi C Ox and

f(X)CP,U{B}. An example of a space Tn(X) with the properties a)-d) is the ordinary `hedgehog' J,, with n `needles'. This example is important in the sequel, but in concrete situations

we may prefer other T. Below, n E N+, X, and Tn are fixed. Simultaneously with X we consider the space CC(X,Tn) of all continuous maps from X to Tn, in the topology of pointwise convergence.

We take the subspace Z = {(x1,... ,xn) E Xn: xi 0 xj if i 0 j} of Xn. Let A C Z. For each x = (XI, ... , xn) E A we choose a neighborhood G,, of x in X" x Ox,,, Where the Ox; are open in X and Ox; n ox, = 0 of the form Cx = Ox, x

ifiq`j.

Properties a)-d) of Tn allow us to fix a continuous map f,.: X - Tn such that fx(xi) E Pi and f.-(Pi) C Oxi, i = I...... . Put U(r) = {g E Cp(X,7;,): g(xi) E I;}. Since the P, are. open in 7;,, U(:r) is open in C,,(X,T,,). Moreover, f., E U(x). Using this notation we have the following simple assertions.

11.5.13. Lemma. If x, y E A and fy E U(x), then x E C.

Proof. U(x) D f, implies that fy(xi) E Pi, i = 1,...,n. But fy'(Pi) C Oyi by the x Oyn = C. definition of fy. Thus, xi E Oyi and x = (x11. .. , xn) E Oy, x 11.5.14. Lemma. Let x, y E A and either x

Gy or y 0 Gz. Then fz # fy.

Proof. Suppose fx = fy. Then U(x) D fs = fy and by lemma 5.12, x E Gy. Similarly, U(y) D fy = f=, hence y e G.. We define the map 7b: A -+ Y = { fx: x E A) by '(x) = f=, x E A. Lemma 5.13 implies

11.5.15. Proposition.

If for every two distinct x, y E A either x 0 Gy or y 0 G,,

then i/i: A ---f Y is bijective.

We consider a concrete situation.

11.5.16. Proposition. If Gx fl A = {x} for all x E A, then U(x) fl Y = { fx}, and hence the space Y is discrete in itself.

Proof. Let y E A and y

x. The condition x ¢ Gy implies, by lemma 5.12, that fy V U(x). Hence U(x) fl Y = { f2}.

11.5.17. Proposition. Always, s(Z) < r = s(C,,(X,T,,)).

5. HEREDITARY SEPARABILITY, SPREAD, AND HEREDITARY LINDELOF NUMBER

71

Proof. Let A C Z and A discrete (in itself). For each x = (x1,...,x,,) E A we can choose open sets OxI,... , Ox" in X such that Ox; n Oxj = 0 if i 0 j, and (OxI x ... x Ox") n A = {x}. Put Gz = Ox1 x

x Ox".

The second construction gives us a subspace Y = { fx: x E A} of C'p(X,T"), with the same cardinality as A (by proposition 5.14) and discrete in itself (by proposition 5.15).

We obtain IA[ < [YJ < r.

Let T" = J be the 'hedgehog' of n intervals emanating from a common point. Conditions a)-d) are clearly satisfied. We have

11.5.18. Theorem. For every n E N+, s(X") < s(Cp(X, J")).

Proof. Induction with respect to n E N+. For n. = 1 we have X = X" = Z, and by proposition 5.16, s(X) := s(Z) < s(Cp(X,JI)). Let k E N+ be such that s(Xk) S s(C'p(X,,1k)). We show that, s(Xk+1) < S(Cp(X,Jkt1)) W e have X'" = 7 . U { I ; j: i, j E { 1,...,k+ 1 } } , where I j = {:r. -_ (.r.1.... , .rk I I) E X'": :r1 :rj }. l'ol' i / 3, every I'; j is llonme0ntorphic (as R slllispawe of X'") to Xk. In fact, the ho neolnorpllisnl between Eli and Xk consists of deleting the ith coordinate in x =---(XI, , xk+1) E F,j (for i t j). Moreover, s(I;j) = s(XL). By assumption, s(Xc) < s(Cp(X, Jk)). Clearly, Cp(X, Jk) is a subspace of Cp(X, Jk+I). For i., j E {1,. - -, k + 1 } we obtain s(F,j) = s(Xk) < s(C'(X, Jk)) < s(Cp(X, Jk+1)) Thus,

s(Xk+1) < s(Z), and proposition 5.16 now implies that s(Xk+l) 5 s(Cp(X,Jk+I)). By induction we conclude: s(X") < s(Cp(X, J")) for all n E N+. Clearly, the 'hedgehog' J2 is homeomorphic to the interval I. Therefore theo-

rem 5.17 for n = 2 implies that s(X2) < s(Cp(X, 1)). But Cp(X, I) C C,(X) _ Cp(X, R). Hence we have

11.5.19. Corollary. Always, s(X2) <s(Cp(X)). We denote by MA + -CH the combination of the Martin axiom and the negation of the continuum hypothesis [3].

It is well known that MA + -'CH is consistent with the ordinary axiom system ZFC of set theory [3].

11.5.20. Theorem.

Assume MA + -NCH. Then every compactum X for which s(Cp(X)) < o is metrizable. Proof. By corollary 5.18, s(X2) < lZo. It has been shown by Senmiklosh that MA + -NCH implies that the compactum X 2 is perfectly normal. Moreover, the diagonal A = {(x, x): x E X) is of type Gs in X x X. Hence X is metrizable [.16). This assertion can be easily generalized as follows:

11.5.21. Proposition. Assume MA+-CH. If s(CC(X)) < lto, then every compact set F lying in X is metrizable.

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H. DUALITY BETWEEN INVARIANTS OF LINDELOF NUMBER AND TIGHTNESS TYPE

Proof. The space CC(X) can be continuously mapped onto CC(F) (see chapt. 0, §4). Hence s(Cc(F)) < s(Cc(X)) < No. It remains to apply theorem 5.19.

Is the assumption that for every compactum X we have Question. w(X) = s(CC(X)) consistent with ZFC? 11.5.22.

Clearly, every `hedgehog' J,, can be represented as a subspace of the plane R2. Thus, CP(X, J") C CC(X, R2).

But Cp(X,R2) is homeomorphic to CC(X) x Cp(X).

Hence

s((Cp(X))2) for all n E N+. This and theorem 5.17 imply

11.5.23. Corollary. s(X") < s(Cp(X) x Cp(X)) for all n E

N+.

11.5.24. Corollary. Always, s(CpCp(X)) = sup{s(X"): n E N+}. Proof. Theorem 5.17 implies that sup{s(X"): n. E N+} > sup{s((C'(X))"): n E N+} > sup{s((CCCC(X))"): n E N+} > s(CpCp(X)).

On the other hand, 5.22 and 5.18 imply that s(X") < s(CC(X) x CC(X)) < s(CCCp(X)).

We will now obtain similar results for the invariants hl and hd. Using the notation from the second construction, we have

11.5.25. Proposition.

If A is a left (right) subspace of a space Z, then CC(X,T") contains a right (respectively, left) subspace Y which has the same cardinality as A.

Proof. Let < be the left (right) well order on A. For x E A we put A,, = {y: y < x} (respectively, A. = {y: x < y}). The set Ay is a left (right) ray in A, hence a closed set in A. Consequently, there is an open set G2 in Z such that G= x = (xl, ... , x")

and Gx = Oxl x ... x Ox", where Oxi fl Ox, = 0 for i # j, and Gx C Z \ Ax. By the second construction we construct the set Y = j f2- x E A). The sets Gx, X E A, satisfy the conditions of lemma 5.13 (since y 0 G,, (x Gy) if y < x). Therefore the map 7/1: A -+ Y is bijective, and it remains to prove that Y is a right (left) subspace of Cp(X,T"). Put ff < fp if and only if x < y. Reasoning as in the proof of assertion 5.8, we find that U(x) fl Y is contained in every left (respectively, every right) ray containing fi. Hence every left (right) ray is open in Y.

II.5.26. Theorem. For every n E N+: a) hl(X") < hd(Cp(X, J")); b) hd(X") < hl(CC(X, in)).

Proof. Proposition 5.24 clearly implies that hl(Z) < hd(Y) and hd(Z) < hl(Y), where Y = Cp(X, J") and Z C X" is as in the second construction. We now prove formulas a) and b) by induction with respect to n. For n = 1,

5. HEREDITARY SEPARABILITY, SPREAD, AND HEREDITARY LINDELOF NUMBER

73

Z = X", and hence the inequalities in a) and b) hold. Assume that hl(Xk) < hd(Y) and hd(Xk) < hl(Y), and let n = k+ 1. Then Xk+1 = Z U {F;,: i, j E { 1, ... , k+ 111, where F, = {x = (xli xk+l) E X'+: xi = x, }> and i # j. Since for i j i, j E {1,. .. , k + 1 }, the subspace Fi, of the space Xk+I is homeomorphic to Xk (see the proof of assertion 5.17) and the invariants hl and lid do not increase under taking the union of a finite family of sets, we have hl(X") < hl(Z) < hd(Y) and hd(X") < lid(Z) < hl(Y).

,

s

>

11.5.27. Corollary. Always, hl(X2) < hd(C9(X)) and hd(X2) < hl(C(X)).

For all n E N+, hl(X") < hd((Cp(X))2) and hd(X") <

II.5.28. Corollary. hl((Cp(X))2)

Corollary. Always, hd(CpCp(X)) = sup{hd(X"): n E N+} and hl(CpCC(X)) _ {hl(X' ): it E N'}. 11.5.29.

An extremely interesting result has been obtained by N. V. Velieliko 1221. He has proved that for every space X,

hd(Cp(X)) = hd(C'(X) x Cp(X)).

This implies, of course, that hd(Cp(X)) = hd((Cp(X))") for all n E N+. We now prove Velichko's theorem.

A set Y C C(X) is called finitely separating if for each finite A C X there is an f E Y such that If(A)I _ JAS. Recall that a set Y C C(X) is called a generating set of functions if it determines the topology of X in the sense that the family {f -'(V): f E Y, V open in R} is a base of the topology (see chapt. 0). We say that a set. Y C C(X) is finitely generating if it is finitely separating and for any finite set A and neighborhood W of A there are a function f E Y and an open set V in R such that

AC f-'(V) CW. An elementary open set W = jj{W: i = 1, ... , n} will be called simple if

0ifi#j.

We begin with a number of lemmas.

11.5.30. Lemma. Let Y C CC(X) and Y a generating set. Then hl (X) < hd(Y).

Proof. Fix a countable base 13 = {V": n E N+} in R and consider an arbitrary family ,y of open sets in X. For each x E U-y we choose Hx E y, fx E Y, and V"(.) E 13

such that x E fi'(V"(x)) C H. Put B" = {x E U-y: n(x) = n} for n E N+, and fix, for each n E N+, a set Bn C B,, of cardinality < hd(Y) such that { fs: x E B;,} is everywhere dense in the subspace {f.: x E B"}. We show that in this case U{Hx : X E U{B, : n r= N+}} = U{Hx : x E Uy}.

(1)

Let x' E Uy and n(x') = n. There is a y E B; such that f5(x') E V", since f..(x') E V". Then x' E f; (V") C Hi,. Equation (1) has been proved. Since U {B,,: n E N+}1 < hd(Y), we have established that any family y of open sets in X

74

It. DUALITY BETWEEN INVARIANTS OF LINDELOF NUMBER AND TICHTNES.S TYPE

contains a subfamily' of cardinality < hd(Y) with the same union: Uy = Uy. Hence hl(X) < hd(Y).

11.5.31. Lemma.

Let Y C C(X) and Y a finitely generating set. Then for every simple set A C Xn the inequality hl (A) < hd(Y) holds. Proof. Let A = 11{ W;:- i = 1, ... , n} C Xn and W; fl W, = 0 for i 0 j. Consider an arbitrary family -y of sets, open in X and lying inside A. The base 13 is assumed to be closed under taking finite unions. Since Y is finitely generating, we can choose for each point x = (XI, ... , xn) E Uy a standard neighborhood W. = f 1{Wx,:

i = 1,... , n} inscribed in y, a function fx E Y, and a set Vn(s) E B such that {XI, ... , xn} C fi' (Vn(y)) C U{ Wx;: i = 1, ... , n}. Reasoning further as in the proof of lemma 2.9 (replacing x by xi,...,x, ), we obtain the required inequality.

A set of the form W(f,V,...,Vn) =

i = 1,...,n} C Xn, where the V

are open in R, is called an R-set in Xn. Here, W (f, V1,. .. , Vn) is a simple R: set if

and only if vi flvi =ofor i/. j. 11.5.32. Lemma. Let V1,... , Vn be a given finite collection of open sets in R, and let {W (f, V 1 ,- .. , Vn) : f E Y} be a family of non-empty R-sets. If Yo is an everywhere dense. set in Y, then

U{W(f,Vi,...,V.) : .f E Yo} = U{W(f,V1,...,Vn) : f E Y}.

(2)

Proof. Let x = (XI, (x1,... , xn) E W (f, VI,... , Vn) for some f E Y. Then there is a g E Yo such that g(x;) E V for i = 1,... , n. We have x; E g- (V ), and hence x E W (g, V1, ... , Vn), which proves (2).

11.5.33. Lemma. Let Y C C(X) and Y a finitely generating set. Then hl(Xn) < hd(Y) for all n E N+. Proof. We prove this by induction. By lemma 5.29, hl(X) < hd(Y). We assume that hl(X'-') < hd(Y), and prove that hl(Xn) < hd(Y). Take an arbitrary set A C Xn and put An = {x = (XI, ... , xn) E A: xi x,, if i 0 j}. Clearly, the set A\ A, is contained in the union of a finite family of sets which are homeomorphic to Xn-'. By induction, this implies that hl(A \ An) < hd(Y). It remains to prove that hl(An) < hd(Y).

Fix a countable base 5 = {Vn: n E N+} in R. For each f E Y and any finite collection

_ (k1,... , kn) of elements of N+ we take an R-set W(f, C) = fI{ f-' (Vk;):

i = 1,...,n}. For each point x E An we now fix some simple R-neighborhood W (fx, &x) of it (this is possible because Y is a finite generating set). By lemma 5.31, for any finite collection e _ (k1,. .. , kn) there is a set Bn, f C An,{ = {x E An: , = f } such

that IBn.d < hd(Y) and U{W(f t:x): x E Bn,{} = U{W (fx, e ): x E An,{}. We denote by Bn the union of the sets Bn,4 over all possible collections C. Clearly, IBnj <

hd(Y) and An C U{W(fx,e ): X E An} = U{W(fx,G): X E Bn}. By lemma 5.30,

5. HEREDITARY SEPARABILITY, SPREAD, AND HEREDITARY LINDELOF NUMBER

75

hd(Y) for all z E A,,, and thus for all x E B. This implies that hl (A") < I B"l hd(Y) < hd(Y). Lemma 5.32 has been proved. We now state and prove N. V. Velichko's theorem.

11.5.34. Theorem. For every space X we have hd(Cp(X)) = sup{hl(X") : n E N+}.

Proof. By theorem 5.10, hd((p(X)) < sup{hl(X"): n E N+}. Since Cp(X) is a finitely generating set, lemma 5.32 implies that sup{hl(X"): n E N+} < hd(Cp(X)). The theorem has been proved. Theorems 5.33 and 5.10 imply

11.5.33'. Corollary. For every space X, hd(Cp(X )) = hd((Cp(X)") for all n E N+. In particular, hd(Cp(X)) = hd(c P(X) x Cp(X)). However, there remain open questions.

11.5.35. Problem. Is it true that hl(Cp(X)) = hl(Cp(X) x Cp(X)) for every space X?

11.5.36. Problem. Is it true that s(Cp(X)) = s(Cc(X) x Cc(X)) for every space X? A positive answer to these questions would allow us to clarify when a space X is zero-dimensional; moreover, this could then be done by a simple change of previous reasonings.

Namely, take for T" the discrete space D" If indX = 0, i.e. X has a base of open-closed sets, then for X and T" = D" the conditions a)-d) hold, with

Pi={i},i=1,...,n,and0=0.

In fact, we only have to verify condition d). Fix a point x E X and a neighborhood

Ox of it. Since ind X = 0, there is an open-closed neighborhood 01x of x such

that 01x C Ox. Let i E

and let the map f : X - T. be defined by: f (y) = i if y E 01x, and f (y) = 0 if y E X \ O1x. Clearly, f is continuous, and

f-'(Pi)=01xCOx.

Reasoning now as in the proof of theorem 5.17, we arrive at the following conclusion:

for all n E N+, s(X") < s(CG(X, D")). But D" C N C R, where N = N+ U {0}. Thus, s(X") < s(Cp(X, N)) < s(C,(X)). Thus we have proved

11.5.37. Theorem.

If ind X == 0, then s(X") < s(Cp(X)) for all n E

Theorem 5.36 implies (see also 1141)

11.5.38. Theorem. If indX

0, then s(CC(X)) = s((CC(X))'I°)

N+.

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If. DUALITY BETWEEN INVARIANTS OF LINDELOF NUMBER AND TIGHTNESS TYPE

We similarly have: if indX = 0, then hl (X") < hd(CD(X)) and hd(X') < hl(C,(X)) for all n E N+. Whence (see [14]), 11.5.39. Theorem. If indX = 0, then h1(Cp(X)) = h1((Cp(X))"0) and hd(Cp(X)) = hd((CC(X))K°).

An important situation in which problems 5.34 and 5.35 have positive solutions was given by M. 0. Asanov [18]. He proved that if the diagonal A of X in its square X x X is of type Gr, i.e. is the intersection of a family y of cardinality < r of open sets in X x X, then s(CC(X)) < r implies that s((CC(X))") < r for all n E N+, while hl(CC(X)) <,r implies that hl((Cp(X))') < r for all n E N+. In particular, if X is a space with G6 diagonal, then s(Cp(X)) = s((CC(X))") and hl(CP(X)) = for all n E N+.

6. Monolithic and stable spaces in Cp-duality A space X is called r-monolithic if nw(A) < r for every A C X such that JAI < r. In particular, a space X is lQo-monolithic if the closure of every countable set in X is a space with a countable network. A space X is called monolithic if it is r-monolithic for every cardinal T, i.e. if for every Y C X we have d(Y) = nw(Y) [13). A separable space of uncountable network weight (in particular, the `arrow' space) is an example of a space that is not No-monolithic.

11.6.1. Examples. The following spaces are monolithic: a) metrizable spaces; b) spaces with a countable network; c) E-products of spaces with a countable base. A space X is called r-stable if for every continuous image Y of X the following conditions are equivalent:

a) iw(Y) < r; b) nw(Y) < r. It is well known that iw(Y) < nw(Y) for every space Y. The converse does not always hold, an example of this being the well-known `arrow' space. A discrete space of cardinality c is not no-stable: it can be condensed onto the

space R, which has a countable base. A space X is called stable if it is r-stable for every infinite cardinal r. It can be easily seen that X is stable if and only if for every continuous image Y of X we have iw(Y) = nw(Y).

11.6.2. Proposition. a) Every compactum is stable; b) Lindelof p-spaces [16] are stable; c) pseudocompact spaces are Ato-stable.

6. MONOLITHIC AND STABLE SPACES IN Cp-DUALITY

77

Proof. Assertion a) is obvious. A proof of an assertion more general than b) will be given later. We prove c). Let X he pseudocompact and f : X -+ Y a continuous map `onto'. Then Y is pseudocompact, and if g: Y --a Z a condensation, where w(Z) = No, then g is a homeomorphism [16]. Thus, nw(Y) < w(Y) = w(Z) = lto. We note the following obvious fact.

II.6.3. Proposition. Every space X is r-stable for every r > nw(X). This and 6.2b) imply that the space of all ordinals smaller than the first uncountable ordinal, in the usual order topology, is stable.

II.6.4. Proposition. Let f : X --> Y be a continuous map from a space X onto a space Y. If X is r-stable (stable), then Y is r-stable (stable). Indeed, every continuous image Z of Y is also a continuous image of X.

11.6.5. Proposition. a) The property of (r-) monolithicity is inherited by arbitrary subspaces; b) stability is inherited by open-closed subspaces.

Proof. Assertion a) immediately follows from the definition and the fact that the network weight is a hereditary cardinal invariant. b) Let X be a stable space, and let Y # 0 be an open-closed subspace of X. Define

the map f:X -+ Y by: f(y)=yforallyeY,andf(x)=yo for all where yo E Y is fixed. Clearly, f continuously maps X onto Y. It remains to apply 6.4.

11.6.6. Proposition.

If X = U(X.: a E A}, where JAI < r and every X. is

T-stable, then X is 'r-stable. This can be proved by a straightforward reasoning.

II.6.7. Corollary. A space which is the union of a countable set of stable subspaces of it is itself stable.

At first glance, monolithicity and stability appear to be rather unrelated to each other. The following theorem indicates that there is a strong relation between them: Cp-duality.

11.6.8. Theorem (A. V. Arkhangel'skii [9], [741).

The space CC(X) is r-

monolithic if and only if the space X is T-stable.

Proof. Necessity. Let Y be the image of X under a continuous map, and iw(Y) 5,r. The space C,,(Y) is homeomorphic to a subspace of Cp(X), hence r-monolithic. In

11. DUALITY BETWEEN INVARIANTS OF LINDELOF NUMBER AND TIGHTNESS TYPE

78

particular, nw(CC(Y)) < d(C,(Y)). But iw(Y) = d(Cp(Y)) and nw(Cp(Y)) = nw(Y). Hence, nw(Y) < iw(Y). Sufficiency. Let M C Cp(X) and IMI < T. Consider the diagonal product map

f = AM of maps from M and put Y = f (X). Thus, f (x) = {x9 = g(x): g E M} and Y is a subspace of Rag. Clearly, w(Y) < IMI < T. We denote by k the set of points in the space Y, endowed with the real quotient topology, corresponding to the map f (see chapt. 0). The identity map is Y - Y is a condensation. Hence iw(Y) < w(Y) < T. Since X is T-stable and can be continuously mapped onto k, we have nw(Y) < T. Further, nw(Cp(Y)) = nw(Y) < T. The space Cp(Y) is homeomorphic to the closed subspace F = {gf: g E CC(Y)} of Cp(X'), since the map f = i-i o f : X -+ Y is a real quotient map (see chapt. 0).

Let g E M. Then g = p9 o f = p9 o i o f, where p9: R°l - R is projection: p9(x) = x9 = g(x). Clearly, the map p9 o is Y --+ R is continuous, i.e. p9 o i E Cp(Y).

Thus g E F, i.e. M C F. Hence M C F = F and nw(M) < nw(F) = nw(CC(Y)) < T. The following theorem reveals that the duality between T-monolithicity and Tstability is two-sided: it sustains reflection.

11.6.9. Theorem (A. V. Arkhangel'skii [9], [74] ).

The space Cp(X) is -r-stable

if and only if the space X is 7--monolithic.

Proof. Necessity. By the previous theorem, the space CpCp(X) is T-monolithic. But X C CCC(X). Thus X is also T-monolithic (for r-monolithicity is inherited by subspaces). Sufficiency. By theorem 6.8 it suffices to derive from the r-monolithicity of X that

CDCp(X) is -r-monolithic. Let M C CCP(X) and EMI < r.

For each f E M we fix, by proposition 2.3 in chapt. 0, a set B f C X such that IBf) < T, while if 81,92 E C,(X) and 911B, = g2IB,, then f(gl) = f(g2). We put A = U{Bf: f E M} and F = A. Clearly, IAI < T. Since X is T-monolithic, nw(F) < T and, hence, nw(Cp(F)) < T (see I.1.1). Consider the restriction map 7r: CC(X) -+ Z C CC(F) (where 7r(g) = 91F and Z = 7r(Cp(X))). Since F is closed in X, the map n: CC(X) Z is open (see

chapt. 0). By the definition of F, for each f E M there is a function hf: Z -+ R such that h f or = f . Because it is a quotient map, h f is continuous, i.e. h f E Cc(Z).

Therefore M C H = {h o 7r: h E CC(Z)I. But II = irO(Cp(Z)) is a closed set in CpC,,(X) since it is a quotient map and H is homeomorphic to CC(Z). We have nw(Cp(Z)) = nw(Z) < nw(Cp(F)) :5,r. Hence also nw(M) < nw(H) < nw(CC(Z)) < T, i.e. CpCp(X) is T-monolithic.

II.6.10. Corollary.

Monolithicity of either X or Cp(X) implies stability of the

other.

11.6.11. Corollary. monolithic (stable).

A space X is monolithic (stable) if and only if CpCp(X) is

6. MONOLITHIC: AND STABLE SPACES IN Cp-DUALITY

79

II.6.12. Corollary.

Every monolithic (stable) space can he embedded. as a closed subspace in a monolithic (stable) linear topological space.

11.6.13. Example. The union of two monolithic spaces need not be monolithic. The Nemytskii plane [161 may serve as an example. It is separable but has no countable network, and hence is not At the same time it is the union of two metrizable (hence monolithic) subspaces, one of which is closed while the other is an open set of type r,,,. 8o-monolithic.

The same example shows that. a space that can be represented as the union of a countable family of closed monolithic (even, metrizable) subspaces need not he 1 omonolithic. The following proposition is of some interest in this respect.

If a space X can be covered by a locally finite family 11.6.14. Proposition. y = {X&: a E Al of closed monolithic subspaces Xa of it, then it is monolithic.

Proof. Let M C X be infinite. Put Ma = M fl Xa. Then nw(M0) < JA1.1 < IMO, and I{a E A: Ma # 0}1 < EMI because -y is locally finite. Therefore nw(U{Ma: a E A}) < EMI. But M= {Ma: a E A}, since y is locally finite.

11.6.15. Corollary. If a space X is monolithic, then the free sum of arbitrary many copies of X (i.e. the product of X by an arbitrary discrete space) is a monolithic space. Assertion 1.6.15 also follows from the next proposition, whose proof is obvious.

II.6.16. Proposition.

The product of a countable family of monolithic spaces is

monolithic.

Theorem 6.9 and corollary 6.15 imply the following, unexpected, result.

11.6.17. Theorem.

If a space Cc(X) is stable, then the space (Cp(X))r is stable

for every cardinal T.

If a space X consists of a single point, then Cp(X) = R is a stable space. Applying theorem 6.17 we obtain

II.6.18. Corollary. The space RT is stable for every T. Besides, this also follows directly from theorem 6.9, since every discrete space X is

monolithic, and Cp(X) = R' for it. Note that every compactum is stable, while every pseudocompact space is No-stable. Therefore we have

11.6.19. Corollary.

If X is a pseudocompactum, then CC(X) is

8o-monolithic.

If

80

II. DUALITY BETWEEN INVARIANTS OF LINDELOF NUMBER AND TIGHTNESS TYPE

X is a compactum, then C,(X) is monolithic. The space T-(wl) of transfinite ordinals smaller than the first uncountable transfinite ordinal is stable. Hence Cp(T-(w1)) is monolithic.

11.6.20. Examples. a. There exists [16] a countably compact space X whose square contains an open-closed uncountable discrete subspace Y. Then X is no-stable, while

Y is not. Hence (see 6.5b)) the space X x X is not No-stable, i.e. No-stability is not preserved, in general, under transition to the square of a space. It is thus even more astonishing that for spaces of the form CC(X) lZo-stability is preserved under transition to an arbitrary power. b. A discrete space M of cardinality c = 21° is not no-stable: it can be condensed onto a space with a countable base, but has uncountable network weight itself. However, M is homeomorphic to a closed subspace of R`. Thus, stability is not inherited, in general, by closed subspaces.

Recall that a space X is a Lindelof p-space if it can be perfectly mapped onto a space with a countable base [16]. The continuous images of Lindelof p-spaces are Lindelof E-spaces. Lindelof E-spaces were introduced by K. Nagami [123). They form

a class of spaces that is remarkable in many respects. In particular, the product of an arbitrary countable family of Lindelof E-spaces is a Lindelof E-space.

11.6.21. Theorem. Every Lindelof E-space is stable. In the proof of this theorem we need the following lemma.

11.6.22. Lemma.

Let f : X --+ Y, g: X -+ Z, and 0: Z - T be continuous

maps, with f (X) = Y, g(X) = Z, and q5(Z) = T. Let, moreover, f be perfect and q bijective. Then Z is the image under a continuous map of a closed subspace of the space Y x T.

Proof of the lemma. Put ib = ¢ o g: X

T and consider zb = iii f : X

TXY

and g' = gZ f : X --+ Z x Y, the diagonal products. The maps ?fi' and g' are perfect, since f is perfect [16]. Clearly, -0' = p o where p: Z x Y -+ T x Y is defined by p(z,y) = (-O(z),y). We denote by it the restriction of p to the set g'(X) C Z x Y.

Then sb = it o g' and the continuity of g' and perfectness of i/i' imply that it is a closed map. Since ¢ is a condensation, it is continuous and bijective. Hence it homeomorphically maps g'(X) onto ?P'(X). Since 0* is closed, tp'(X) is closed in T x Y. Clearly; Z is the image of the space g*(X) under the projection Z x Y -+ Z (since g(X) = Z). Consequently, Z can be represented also as the image of the space 1,*(X) (which is closed in T x Y) under a continuous map.

Proof of theorem 6.21. Let Z be a Lindelof E-space. Since the image of a Lindelof E-space under a continuous map is clearly a Lindelof E-space, it suffices to prove that

6. MONOLITHIC. AND STABLE SPACES IN Cp-DUALITY

81

for the space Z itself the network weight does not exceed the i-weight. So, let r = iw(Z) and 0: Z --. T a condensation, where the weight of T does not exceed r. Since Z is a Lindelof E-space, there are a space Y with a countable base,

a space X, a perfect map f : X - Y, and a continuous map g: X -+ Z such that f (X) = Y and g(X) = Z. The maps f, g, 0 satisfy the conditions of lemma 6.22. Thus, Z is the continuous image of a (closed) subspace of the space Y x T. Whence we conclude: nw(Z) < nw(Y x T). But w(Y) _< No and w(T) < r. Thus nw(Y x T) < r and nw(Z) < r, as required.

We now prove that the class of stable spaces is considerably larger than could be concluded from the results given above. We need the following factorization theorem, which was proved, in essence, by R. Engelking in 166].

11.6.23. Theorem. Let X = fj{Xa: a E A} be a topological product, T = a fl{Xa: a E All the a-product of spaces over a point x' E X, T C A C X, r an infinite cardinal, and suppose that for each finite set K C At the Lindelof number of the space XK = fI(Xa: a E K) does not exceed r (i.e. l(XK) < r). If f : A - Y is a continuous map and iw(Y) < r, then there is a set L C Al such that I LI < r, while if x,x' E A with xa = x'a for all a E L, then f (x) = f (x').

11.6.24. Proposition.

Let A. be a subspace of a product X = n{Xa: a E All, f a realvalued continuous function on A, and L, L C M, a set such that the following two conditions hold:

a) if x, x' E A and xa = x'a for all a E L, then f (x) = f (x'); b) the map PLIA: A PL(A) C fi{Xa: a E L} is a quotient map. Then there is a uniquely defined continuous map q : pL(A) - R for which f IA = ,, o (pLIA)

Proof. a) implies that a map 0: PL(A) -+ R for which PAA = q, o (PLIA) exists and is unique. Since PLIA: A - pL(.A) is a quotient map and the map f IA is continuous, the relation PAA = 0 o (pLIA) implies that 0 is continuous.

11.6.25. Theorem. Let X = I1{Xa: a E M} be a topological product, T = a fI{Xa: a E M} the a-product of spaces over a point x' = Ix,*,: a E M} E X, and T C A C X. Let, moreover, the following conditions hold (for a fixed r > No): a) for every finite K C M the Lindelof number of the space XK = n{Xa: a E K} does not exceed r (i.e. l(XK) < r); b) if L C M and I LI < r, then the space pL(A) C XL = fI{Xa: a E L} is r-stable and the map PLIA: A -+ PL(A) is a quotient map.

If f : A -+ Y is a continuous map with f (A) = Y, then Y is a r-stable space. In particular, A is itself r-stable. The two main consequences of this theorem relate to the cases when A = X and A = T (here PL is an open quotient map).

82

If. DUALITY BETWEEN INVARIANTS OF LINDELOF NUMBER AND TIGHTNESS TYPE

Proof of theorem 6.25. By proposition 6.4 we only have to prove that A is r-stable. Let g: A -+ Y be a continuous map, g(A) = Y, and iw(Y) < r. Clearly, the assumptions of theorem 6.23 are fulfilled. Thus there exists an L C M with ILI < r and such that if x, x' E A and xa = x'' for all a E L, then g(x) = g(x'). Since the map PLIA: A -+ pL(A) is a quotient map, we can now apply proposition 6.24. We conclude: there is a continuous map 0: pL(A) --+ Y such that ¢b(pL(A)) = g(A) _ Y.

By assumption, pL(A) is r-stable. Hence iw(Y) < r implies nw(Y) :5,r. Hence A is r-stable. The assumptions of theorem 6.25 are fulfilled if the weight of every X., does not exceed r, and A = X or A = T. In particular, we obtain

11.6.26. Corollary.

The product, E-product, and a-product of an arbitrary family of spaces with a countable base are stable spaces.

Theorem 6.21 allows us to apply theorem 6.25 to the class of Lindelof E-spaces, since the product and a-product of a countable family of Lindelof E-spaces are Lindelof Espaces [13). We obtain

11.6.27. Corollary.

The product, E-product, and a-product of an arbitrary family of Lindelof E-spaces are stable spaces.

Note that the E-product of an arbitrary set of space with a countable base is simultaneously a stable and a monolithic space (see 6.26 and 6.1).

11.6.28. Theorem. Every Lindelof P-space X is

No-stable.

Proof. Let the maps f : X --+ Z and'g: Z --+ T be continuous, f (X) = Z, g bijective, and let T be a space with a countable base. We show that Z is countable. Since g is a condensation, every point in Z is of type G6. The continuity of f and X being a P-space imply that the set f-1(z) is open in X for every z E Z. The elements of the open cover ry = If-'(z): z E Z} of X are pairwise disjoint. Since X is a Lindelof space, -y is countable. Hence IZI = I7I < No. The product of finitely many Lindelof P-spaces is a Lindelof P-space. The proof of the following lemma is obvious.

11.6.29. Lemma.

The a-product of a countable family of spaces is covered by a

countable family of subspaces each of which is homeomorphic to the product of a finite number of factors.

Lemma 6.29, the remark preceding it, and the fact that the union of a countable family of lto-stable spaces is an No-stable space (proposition 6.6) imply that if X is the or-product of a countable family of Lindelof P-spaces, then X is an l 0-stable space. Applying theorem 6.25 we arrive at the following result.

7. STRONG MONOLITHICITY AND SIMPLICITY

11.6.30. Corollary.

83

The a-product of any set of Lindelof P-spaces is an loo-stable

space.

Theorems 6.8 and 6.9 imply that stability and monolithicity are supertopological properties. Moreover, by combining these theorems we obtain the following corollaries.

11.6.31. Corollary.

If Xr % Y for some n, and X is stable (monolithic), then Y is

stable (monolithic).

11.6.32. Corollary.

For any X the following conditions are equivalent: a) X is stable (monolithic); b) Cp,2 (X) is stable (monolithic) for all n E N+. Our understanding of the class of No-stable spaces is enriched in another direction by the following result.

11.6.33. Theorem.

If the Ilc:oitt-Nachbin rralcompactification vX of a space X

is an No-stable space, then the slace X is itself lQo-stable.

Proof. Let f : X - Y be a continuous map and g: Y -- Z a condensation, where Z is a space with a countable base. We must prove that the network weight of Y is countable. Note that Z and Y are realcomplete spaces. In fact, every space that can be condensed onto a Lindelof space of countable pseudocharacter is realcomplete [66).

Thus, vY = Y and vZ = Z. Consider an extension f : vX - vY" = Y of f. Since vX is

No-stable,

the space Y has a countable network. This proves the theorem.

11.6.34. Corollary. If vX is a Lindelof E-space, then X is lZo-stable.

11.6.35. Corollary. If vX is homeomorphic to a product of Lindelof E-spaces, then X is lto-stable.

7. Strong imonolithicity and simplicity The material of this and the previous section are strongly related. Here we consider a strengthening of the notions of r-stability and r-monolithicity, and find a duality between them.

A space X is called strongly r-monolithic if for every A C X with IAA < r the weight of the space -A does not exceed r. We say that X is strongly monolithic if this condition holds for all r > No [11.). Of course, every strongly monolithic space is monolithic. All metric spaces are strongly monolithic. A space with a countable network but without countable base is monolithic, but not strongly Iio-monolithic.

11.7.1. Proposition. A compactum X is strongly r-monolithic if and only if it is r-monolithic.

84

11. DUALITY BETWEEN INVARIANTS OF LINDELOF NUMBER AND TIGHTNESS TYPE

Proof. It suffices to note that for a compactum the network weight and the weight coincide [116].

A space X is called r-simple if for every continuous map f from it into a space of weight < r the cardinality of the set f (X) does not exceed r. A space is called simple if it is r-simple for all r > No [11]. Of course, an uncountable discrete space is not No-simple, while every countable space is simple. The following four assertions are obvious.

11.7.2. Proposition. The image of a r-simple space under a continuous map is a r-simple space.

11.7.3. Proposition.

Every subspace of a strongly r-monolithic space is strongly

r-monolithic. 11.7.4. Proposition. The product of a. countable family of strongly monolithic spaces is a strongly monolithic space.

11.7.5. Corollary.

The product of a strongly monolithic space and a metric space

is strongly monolithic.

If every set of type GT is open in X, then X is called a PT-space.

11.7.6. Theorem. Every Lindelof P-space is lio-simple. We prove a somewhat stronger assertion. We put ld(X) = sup{I yI: -y is a disjoint open cover of X}, and call the cardinal invariant ld(X) the discrete Lindelof number of X [11]. Clearly, ld(X) never exceeds the Lindelof number of X and the Suslin number of X.

11.7.7. Proposition.

Let X be a PT-space and ld(X) < r. Then X is r-simple.

Moreover, if f : X -+ Y is a continuous map from X into a space Y such that i,b(Y) < r (i.e. every singleton subset {y} C Y is of type GT in Y), then If (X)I < r.

Proof. For each y E Y we fix a family -yy of open sets in Y such that Iryyl < r and

flryy = {y}. Then f-'(y) = fl{f-'(U): U E y} is an open set in X, since X is a P,-space. The family p = { f -' (y): y E Y} is a disjoint open cover of X. Thus, I µI < r. Since f (X) = Y, we conclude that IYI = I µI <_ r. Clearly, theorem 7.6 follows from proposition 7.7. It is well known that if the network weight of a space X does not exceed r, then the space can be condensed onto a space of weight < r (see [3]). Whence,

II.7.8. Proposition. Every r-simple space is r-stable.

7. STRONG MONOLITHICITY AND SIMPLICITY

85

An unexpected fact is the strong relation between strongly monolithicity and simplicity: it is the duality expressed in the following theorem.

11.7.9. Theorem (A. V. Arkhangel'skii [1]). A space X is r-simple if and only if Cp(X) is strongly -r-monolithic.

Proof. Necessity. Let X be T-simple and.T C Cp(X), 1.T1 < r. Consider the diagonal

product ¢ = A.T: X -+ R', q5(x) = {xf: f E F}, where xf = f (x) for x E X. Put Y = ¢(X). Since X is r-simple, 0 is continuous, and w(Y) < -r, we have JY, < T. Clearly, .T C P, where P = {go f: g E R'} is a closed subspace of RX homeomorphic to R1. Consequently, .T C P and w(T) < w(P) < w(R") < FYI < T. Sufficiency. Let 0: X -+ Y be a continuous map and w(Y) < r. Then d(Cp(Y)) < nw(Cp(Y)) < nw(Y) < T. The space Cp(Y) is homeomorphic to some subspace P of Cp(X). Hence Cc(Y) is strongly r-monolithic, and d(Cp(Y)) < r implies that w(Cp(Y)) < T. Thus, FYI < r (in fact, Cp(Y) is everywhere dense in R'', and hence. w(C,,(Y)) > IYI).

11.7,10. Corollary [11].

The space Cp(X) is a strongly lto-monolithic PrechetUrysohn space if and only if the space X is loo-simple and l(X") < No for all n E N+.

Proof. Necessity. If Cp(X) is a FYechet-Urysohn space, then its tightness is countable, and hence 1(X") < No for all n E N+ (see §1). This and theorem 7.9 imply the necessity part. Sufficiency. By theorem 7.9 and results in §1, the space Cp(X) is strongly ftcmonolithic and t(Cp(X)) < No. Let f E C ,,(X), M C Cp(X), and f E M. There is a countable B C M such that f E B. But B is a space with a countable base. Thus there is sequence in B converging to f.

11.7,11. Corollary. Simplicity of a space X is a supertopological property: if Cp(X ) is homeomorphic to CC(Y) and X is T-simple, then Y is r-simple. As distinct from the situation with monolithicity and stability, theorem 7.9 cannot be reflected. Moreover, strong lZo-monolithicity of a space X cannot even be characterized in terms of linear topological properties of Cp(X ): there exists spaces X and Y such that X is strongly monolithic, Cp(X) and Cp(Y) are linearly homeomorphic, and Y is not strongly No-monolithic.

II.7.12. Problem. Let X be r-simple and X-' Y (X-" Y for some n > 2). Is it true that Y is r-simple? Recall that a space is called scattered if every nonempty subspace Y of X contains an isolated (in Y) point.

11.7.13. Theorem. Every scattered Lindelof space is a simple space.

86

H. DUALITY BETWEEN INVARIANTS OF LINDELOF NUMBER AND TIGHTNESS TYPE

Proof. Let (X, T) be a scattered Lindelof space. Fix an arbitrary cardinal T > Ito and let T, be the topology on X whose base is the family {fly: y C T and HryH < r}. We now need the following lemma, which is due to V. V. Uspenskii [75].

11.7.14. Lemma. If the space (X, T) is scattered and its Lindelof number does not exceed r, then the Lindelof number of the space (X, T) does not exceed r.

Proof of the lemma. Let -t C T, and Uy = X. Put £ = { V E T: there is a family p Cry such that V C Up and 1,a1 < r), and Y = U£. The following assertion holds:

*) if Z is a closed set in (X, T) and Z C Y, then there is a y' C 'y such that

fy'1
In fact, 1(X, T) < r and £ C T imply that there is a subfamily E' C E for which 1£*1 < r and UE' J Z. Using the definition of E, we conclude that there is now a subfamily y' Cry for which 17'1 < r and Uy' D Z. If X = Y, a reference to property *) finishes the proof of the lemma. Let X \ Y # 0. Since X is scattered, there is a point x E X \ Y which is isolated in Y. Take U E T such that x E U E U C {x} U Y (closure with respect to (X, T)). There is an A E y for which x E A. Since A E T we can choose a family 0 C T such that 101

For each G E 0 the set U\G is closed in (X,T), and U\G C Y = UE. Hence we can use property *) and choose a subfamily -to c -t such that IycI < r and Uyo 3 U \ G. Put y = U{-yam: G E o} u {A}. It is obvious that HHH < 7 - -1 0 1 = r, ' C y, and Uy D U D x, whence U E £ and x E U£ = Y. This contradicts x Y. The lemma has been proved. We continue with the proof of theorem 7.13. The space (X, T,) is clearly a P,-space, and by lemma 7.14, ld(X, 2) < r. By proposition 7.7 we conclude that the space (X, T,) is r-simple. But (X, T) is a continuous image of (X, T,). This implies (see proposition 7.2) that (X,T) is r-simple. Since r > 1Zo is arbitrary, it follows that (X, T) is simple.

11.7.15. Theorem [13]. Let X be a Lindelof P-space. Then CC(X) is a PrechetUrysohn space, and CC(X) is strongly lZo-monolithic.

Proof. Since X is a Lindelof P-space, the space X" is Lindelof for all n. E N+. By theorem 7.6 the space X is no-simple. Applying corollary 7.10 we conclude that Cp(X) is a strongly lZo-monolithic FrAchet-Urysohn space.

11.7.16. Theorem [13]. If X is a scattered Lindelof space, then CC(X) is a strongly monolithic Frechet-Urysohn space. Proof. Lemma 7.14 implies that X is a continuous image of some Lindelof P-space Y. Thus CC(X) is homeomorphic to a subspace of Cp(Y) (see chapt. 0), and theorem 7.15 implies that Cp(X) is a strongly l-o-monolithic FY-echet-Urysohn space. The theorem has been proved.

8. DISCRETENESS, IS A SUPERTOPOLOGICAL PROPERTY

87

Note that corollary 7.10 remains true if the assertion `C,,(X) is a Frechet-Uryysohn space' is replaced by 'Cc(X) is a k-space', or by `C,(X) is a sequential space'. This follows from results in §3. As we will see in the next chapter, for compacta a converse of theorem 7.16 is true:

if X is a compactum and Cc(X) is a Frechet-Urysohn space, then X is scattered 1171.

8. Discreteness is a supertopological property In this section we give three topological properties of a space C,,(X) which together

correspond to the fact that X is discrete. This implies that a space which is tequivalent to a discrete space is itself discrete (a theorem of V. V. Tkachuk 1531).

Recall that for any set X the space RX has the Baire property, is realcomplete, and stable (see chapt.. ii, §3 and chapt. 11, §6). Invoking theorems 1.3.4, If.(;. 18, And 11.6.10 we thus obtain

II.8.1. Proposition.

If a space. Cp(X) is homeomorphic to R." for some Y, then every bounded set in X is finite, the space X is monolithic, while if X is not discrete, then X contains a countable non-closed set.

As we already know (see chapt. I, §3), there exists a countable nondiscrete space X for which Cc(X) has the Baire property. For this X the space CD(X) is realcomplete and stable (being a space with a countable base), but not Cech complete (see chapt. 1, §3) and hence not homeomorphic to a space R1' for any Y (if Y is uncountable, R}, has a countable base and is hence not homeomorphic to CG(X)). Thus, the presence in Cc(X) of the three properties: the Baire property, realcompleteness, and stability, does not yet imply that X is discrete. Below we give yet another nontrivial property of a space RX, allowing us to solve our problem. A space Y is called projectively complete if every space with a countable base which is the image of Y under a continuous open map is metrizable by a complete metric 110).

11.8.2. Proposition. The space RX is projectively complete.

Proof. Let f : RX

Y be a continuous open map, f (RX) = Y, and Y a space

with a countable base. Then there are a countable set. A C X and a continuous map g: R' Y such that g o 7r = f, where 7r: RX -- RA is projection (see chapt. 0). Since f is open, g is open. But RA is a space with a countable base which is metrizable by a complete metric, since A is countable. By a well-known theorem of Hausdori 1161, Y is then also metrizable by a complete metric. Regrettably, it is yet unclear whether the projective completeness of a space CP(X ) can be characterized in terms of topological properties of X. However, we have the following, for us very important., proposition.

88

II. DUALITY BETWEEN INVARIANTS OF LINDELOF NUMBER AND TIGHTNESS TYPE

11.8.3. Proposition. If C9(X) is projectively complete, then every countable closed subspace F of X is discrete and C-embedded in X.

Proof. Consider the restriction map 0: Cp(X) -' Cp(F) (i.e. ,i(f) = f IF) and put Y is open. Moreover, Y = tI (Cp(X)) C Cp(F). Since F is closed, 10: Cp(X) V = Cp(F). Finally, F is countable and Y C RF. Thus Y is a space with a countable base. Since Cp(X) is projectively complete, we conclude that Y is Cech complete. But then (see chapt. I, §3) Y = Cp(F) = RF, hence F is discrete and C-embedded in X. In the proof of the main result of this section we also need the following

11.8.4. Lemma.

If M is an uncountable space with a countable network, then M contains a convergent sequence {yn: n E N+} such that yn' # yn" if n' # n", i.e. M contains a nondiscrete countable compactum.

Proof. Fix a countable network S in M and put L = {y E M: there is a finite A C S such that {y} = na}. The set L is countable, since S is countable. Thus, M \ L t 0. Fix y' E M \ L and put 9 = {P E S: P 3 y'}. The elements of the countable family 9 can be enumerated: 9= {Pn: n E N+}. n PA, is infinite (otherwise y E L). This allows us to successively The set PI n choose yn E PI n n Pn, for all n E N+, such that yn' t yns. if n' # n". Since S is a network in M and 0 is the family of all elements in S containing y', the sequence {yn: n E N+} converges to y'.

11.8.5. Theorem [10].

If the space Cp(X) is projectively complete, stable, and realcomplete, then X is a discrete space.

Proof. Assume X to be not discrete. Since Cp(X) is realcomplete, X contains a countable nonclosed set A (see §4). Then 7 is a closed nondiscrete subspace of X. The projective completeness of Cp(X) implies, by proposition 8.3, that A is uncountable. But X is monolithic, since Cp(X) is stable (see §6). Thus 7 is a space with a countable network. Lemma 8.4 implies that A contains a nondiscrete countable compactum K. But the countable set K is closed in A and in X. Hence proposition 8.3 implies that K is discrete. We have obtained a contradiction.

11.8.6. Corollary. If Cp(X) is projectively complete, stable, and realcomplete, then Cp(X) = RX.

11.8.7. Corollary (V. V. Tkachuk [53]). The space Cp(X) is homeomorphic to RX if and only if X is discrete.

Proof. By theorem 8.5 it suffices to note that RX is realcomplete, stable, and projectively complete (propositions 8.1 and 8.2). The notion of projective completeness is a sufficiently far-reaching generalization

8. DISCRETENESS IS A SUPERTOPOLOCICAL PROPERTY

89

of the notions of metric completeness and Cech completeness. This is clear from the fact that, e.g., not every projectively complete space has the Baire property 110). On the other hand, not every space with the Baire property is projectively complete (see the discussion of an example at the beginning of this section).

11.8.8. Problem.

Is it true that the space Cp(,6N) is projectively complete? Is it true that the space CC(QN \ N) is projectively complete?

11.8.9. Problem. Let X ^' Y with X discrete. Is then Y discrete?

11.8.10. Problem.

Is the projective completeness of C9(X) characterized by the property that every countable closed subspace of X is C-embedded in X and discrete?

CHAPTER III

Topological properties of function spaces over arbitrary compacta As is clear from the general results already exposed by us, the space Cc(X), where X is an arbitrary compactum, has substantial peculiarities. In particular, all such sp wes are nionolit.hic (see chapt.. 11, §6) and have countable tightness (chapt. 11, §1). The aim

of this chapter is to deepen our understanding of the peculiarities of the topological structure of function spaces over compacta, and to systematically investigate these peculiarities.

1. Tightness type properties of spaces Cp(X), where X is a compactum, and embedding in such Cp(X) Every compactum is a Hurewicz space (see chapt. II, §2). Hence theorem 11.2.2 implies

III.1.1. Theorem. For each compactum X the fan tightness vet(CC(X)) of Cp(X) is countable (moreover, the tightness of Cp(X) is countable). However, if X is a compactum, then Cp(X) is only in exceptional cases a sequential space or a k-space. Indeed, let Cp(X) be a k-space. Then by theorem 11.3.7, Cp(X) is a Frechet-Urysohn space. At the same time, by lemma 11.3.5, Cp(IO,11) is not a FYechet-Urysohn space. Hence (see chapt. 0), X cannot be continuously mapped onto [0, 11. But a compactum that cannot be continuously mapped onto an interval is scattered. Thus we have proved that if X is a compacturn and Cp(X) is a k-space, then X is scattered. However, by theorem 11.7.16, if a compactum X is scattered, then CC(X) is a Ffechet-Urysohn space. We thus have

111.1.2. Theorem [47], [94]. Let X be a compactum. Then the following conditions are equivalent: a) Cp(X) is a Frechet-Urysohn space; b) CC(X) is a k-space; c) the compactum X is scattered.

In particular, among the metrizable compacta only the countable ones have the 91

111. TOPOLOGICAL PROPERTIES OF FUNCTION SPACES

92

property that CC(X) is a Frechet-Urysohn space. Since the topology of a space C,,(X) over a countable compactum is determined by the system of its countable subspaces, but convergent subsequences do not, as a rule, suffice, there arises the following question: what countable spaces can be contained in CP(X) if X is a compactum? Theorem 1.1 implies (see chapt. II, §2) that already the countable Frechet-Urysohn fan V(ldo) is not contained in CC(X) for any compactum X, since vet(V(No)) > 1 o and vet(CC(X)) < 1 o Note that V(ldo) belongs to the class of simplest countable spaces with a unique nonisolated point and having the Frechet-Urysohn property. The central results of this section, theorems 1.6 and 1.10, due to V. V. Uspenskii, give an answer to the above stated question. Below we use the following notations.

We consider a space Z and a fixed point g of it. If F C Z, then XF denotes the characteristic function of F on Z, i.e. XF(z) = 1 if z E F and XF(z) = 0 if z E Z \ F. By V we denote the discrete colon 10, 1), and by D7 the Tikhonov product of Z copies of V. If F C Z, then XF E D7. We put

={FCZ: g¢7},

(1)

W={XF: FEf}CVz.

(2)

We now assume that Z C CC(Y), where Y is a compactum. We take arbitrary n E N+ and k E N+, and define the families 7)n,k and r,,,k of sets in Z as follows. For each = (y1, ... , yk) E Yk we set

Fg={f E Z : max{lf(yi)-g(y;)j : i=1,...,n}> 1/n).

(3)

It is clear that FF is closed in Z and g 0 FD. Thus, Fv E £.

(4)

For arbitrary f E Z and K C Z we put

Pf={yEYk: f EFy},

PK=n{Pf: fEK}.

(5)

It can be readily verified that Pf is closed in Yk. Hence PK is closed in Yk. We consider the following families of sets in Z:

77n,k={F#:9EYk}C

cc

(6)

rln,k = {M C Z : there exists a y E yk for which M C FF}.

(7)

Bn,k = {XF : F E 'nn,k} C W

(8)

We put

111. 1.3. Proposition. Let n, k E N+ be given, L C Z, and let for each finite set K C L there be a ,y" E Yk such that K C FF (i.e. K E n,k) Then there is a y E Yk such that L C Fu, i.e. L E Tln,k.

Proof. The family -y = {PK: K C L, IKI < Ro} consists of nonempty (by the conditions) closed sets in the compactum Yk. Clearly, PK,uK, = PK1 n PK2. Thus the

1. TIGHTNESS TYPE PROPERTIES OF SPACES Cp(X)

93

family ry is centered and P= (lyy 0. For any y E P and f E L we have -Y3 P{f} 3 i.e. f E .P#. Consequently, L C Fy for y E P.

111.1.4. Proposition.

The set B,,,k is closed in the compactum. Vz.

Proof. Let ¢ E Dz and 0 E B,,,k. Then qS(g) = 0, since XF(g) = 0 for all F E rt,,,k. By the definition of Dz there is an M C Z such that 0 = XM. Now Xnr E Bn,k implies that for each finite K C M there is a y E Y'k such that K C Fg. But then, by proposition 1.3, M E n,k, and hence 0 = XM E B,,,k. Proposition 1.4 implies in a simple manner a very important consequence.

Theorem [59]. If g E Z C CC(Y), where Y is a compactum, then W = {XF: F C Z, g V F} is a subspace of type K, of the Tikhonov product Dz of 111.1.5.

(IZI copies of) the discrete colon.

Proof. If F C Z and g V F, then there are n., k E Ni' and g E yk such that

FflIf EZ: If(y2)-g(Y;)I <1/n foralli=1,...,k}=0. Then F C FF, and hence F E ijn,k and Xp E B,,,k. Thus we have

WCU{Bk: n,kEN+}.

(9)

The converse inclusion is trivial (see (8)). This implies that

W=U{B.,k: n,kEN+}

(10)

is the union of countably many compact sets in Dz. Note that the coordinate g E Z can be ignored in considering W as a subspace of Dz, since all f E W vanish at g. In this sense we say that W lies in Dz\{g} If Z is infinite and g is an isolated point in it, the conclusion of theorem 1.5 remains true also without the additional assumptions: W is homeomorphic to the space Dz (more precisely, to Dz\{9}) In the important special case that g is the unique nonisolated point of Z, theorem 1.5

has a converse. In the notation introduced above we have

111.1.6. Theorem [59].

Let all points of Z, except, possibly, g, be isolated in Z. Then the subspace W = {Xp: F (-- Z, g F} of Dz is the union of a countable family of compacta if and only if Z is homeomorphic to a subspace of the space Cp(Y), for some compactum Y.

Proof. By theorem 1.5, we only have to prove necessity. Since all points of Z distinct from g are isolated, all XF with F C Z, g 0 F, are (realvalued) continuous functions on Z, i.e. 14' is a subspace of CC(Z). This same reasoning makes clear that W generates the topology of Z, i.e. the canonical map ?ckw: Z -> CC(W) (see chapt. 0)

is a homeomorphism `into'. But III II < 1 for all f E W, and W = U{F,,: n E N+}, where each F is a compactum. Put F = {(1/n)f: f E and W = U{F,,: n E N+} U { fo}, where fo = 0 on Z. Clearly, W is a compact subspace of the space

111. TOPOLOGICAL PROPERTIES OF FUNCTION SPACES

94

CP(Z) which also generates the topology of Z. The latter means that the canonical map tPw homeomorphically maps Z onto some subspace of CP(W). Using the criteria obtained we will analyze some examples.

111.1.7. Example [591. Consider the Stone-Cech compactification ON of the discrete natural number set N = N+ U {0}. Take an arbitrary point p E ON \ N. We show that the space MM = N U {p} cannot be homeomorphically embedded in the space CC(X), for any compactum X. Assume the contrary. For Z we take the space M, and for g the point p. It is convenient to ignore in

this case the point p in Z (see the remarks above theorem 1.6), and to assume that W lies in DN. The point g = p can be represented by an ultrafilter on N. Then 14' can be described as follows: W = {XF: F C N, and F 0 p). Consider the homeomorphism h from DN onto itself under which in each coordinate 0 is mapped to 1, and 1 to 0.

Since p is an ultrafilter, for every F C N the conditions F o p and N \ F E p are equivalent. Clearly, h(XF) = XN\r for all F C N. These remarks imply that

h.(W) fl W = 0 and h(W) U W = {XF: F C N) = DN. From the assumptions made above, theorem 1.5 now implies that W is a set of type K, in DN. But W does not contain any nonempty open set in VN. This follows from the fact that for any two disjoint finite subset L and K of N we can find a set F C N such that F E p, F 3 L, and F fl K = 0. Thus, W is the union of countably many nowhere dense compacta in DN. Since h is a homeomorphism of DN onto itself, the set h(W) is also the union of countably many nowhere dense compacta in DN. But DN = WUIt (W), and we arrive at a contradiction with Baire's category theorem for compacta.

Example. Put Z = (N x N) U {a}, where a V N x N, declare all points of N x N to be isolated, and take as local base at a the family of sets of the form {(n, k): n > f (k)}, where f : N -> N is an arbitrary map. We obtain the 111.1.8.

well-known Frechet-Urysohn space with unique isolated point a, called the countable Frdchet-Urysohn fan and denoted by V(No) (see chapt. II, §2). We show with the help of theorem 7.5 and bypassing theorem 11.2.2 that Z cannot be embedded in CC(Y) for any compactum Y.

Consider the subspaces Zi = (N x {i}) U {a} of Z. Let W and W1, i E N, be defined from Z and g = a, respectively from Z; and g = a, by formula (2). Clearly, every W; is a-compact. Since a is not isolated in Z1, lei; is not compact. Consequently, there is for each i E N an infinite closed discrete set B, in l'V . The definitions of W and W, clearly imply that W is homeolnorphic to the Tikhonov product fI{ W1: i E N}. This product, however, contains the non-a-compact closed subspace B = f1{B;: i E N}. We conclude: W is not a-compact. Consequently, by theorem 1.5, Z cannot be embedded in CC(Y) for any compactum Y. The fact that not every countable space can be embedded in C, over a compactum can also be derived, bypassing theorem 1.5, from simple cardinality considerations.

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95

Such a reasoning may he as follows, [59]. If a countable space S could be embedded

in Cc(Y), with Y some compactum, then S could be embedded in Cp(Ys), with Ys a metrizable compactum (the image of Y under the diagonal product map from S). But Ys, as every metrizable compactum, is a continuous image of the Cantor perfect set D5o [16, chapt. 3, no. 299]. Hence S and Cp(Ys) could be embedded in Cp(D"0). Since [Cp(D"0)I < 214°, the space Cp(D"°) has at most 2"° distinct countable subspaces. However, there exist 221o pairwise nonhomeomorphic countable spaces.

In particular, this reasoning implies that there is a point p E /3N \ N for which N U {p} cannot he embedded in Cp over a compactum. From theorem 1.5, however, we know that this is true for every point p E ,QN \ N. We call a space Y an Eberlein.-Grothendieck space, or EG-space [2], if it is homeomorphic to a subspace of the space Cp(X) for some compactum X.

Is there an 'intrinsic' topological property that in combiProblem. nation with mmnolithieit;y and rountable fan tightness characterizes the Eberlein -

111.1.9.

Grothendieck Spaces?

We will now discuss to what extent the above results can be generalized to spaces Cc(X) where X belongs to a class larger than the class of compact.a.

If X is a pseudocompact space, then its Hewitt-Nachbin compactification vX and Stone-Cech compactification QX coincide: vX = /3X [16]. By theorem II.4.29, the topologies generated by Cp(X) and Cp(/3 X) on an arbitrary countable set A C C(X) = C(6X) coincide. This implies

111.1.10. Theorem.

If a countable space Y can be embedded as a subspace in Cp(X), where X is a pseudocompact space, then there is a compactum k such that Y is hom.eomorphic to a subspace of Cp(X). In particular, the spaces considered in examples 1.7 and 1.8 cannot be embedded in Cc(X) for any pseudocompact space X. A related result holds for a-compact spaces.

111.1.11. Theorem.

Let X be a a-compact space. Then there is a compactum F such that Cp(X) is homeomorph.ic to a subspace of Cp(F). The proof of this assertion is based on the following lemma, whose proof is obvious.

111.1.12. Lemma.

Let ¢: It - (-1, 1) be a homeomorphism, and let the map

h: Cp(Y) - Cp(Y, (-1, 1)) be defined by: h(f) = q5o f for all f E Cp(Y). Then h. is a homeomorphism of the space C'(Y) onto its subspace Cp(Y, (-1,1)), and the image of any generating (separating) set A C Cc(Y) under h is a generating (separating) set.

Proof of theorem 1.11. Put Y = Cp(X), and let Z be the image of X under the

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canonical map t/ix : X - CpCC(X) = CC(Y) (see chapt. 10). Then Z is homeomorphic to X, o-compact, and generates the topology on Y. By lemma 1.12, the image h(Z)

tinder the homeomorphism h: CC(Y) -+ CC(Y, (-l, 1)) is a v-compact subspace of CC(Y), inducing the topology on Y. We have h(Z) = U{Fi: i E N+}, where the

F, are compacta. Then also, Pi = {(1/i)f: f e Fi} are compact, for all i E N+; moreover, if g E 4i;, then lg(y)l < 1/i for all i E N+. Hence the subspace 4, = U{4,i: i E N+} U {go} of CC(Y) is also compact, where go(y) = 0 for all y E Y. Clearly, the family 4i of functions also generates the topology on Y. Hence the canonical map iiy: Y --' Cp(4)) is a homeomorphism of the space Y = Cp(X) onto a subspace of C,(4?). Theorem 1.11 has been proved. In particular, the spaces in examples 1.7 and 1.8 cannot be embedded in CC(X) if X is a o-compactum. It is now natural to consider which spaces can be realized in CC(X), where X is a space with a countable base. This problem is solved by the following result of V. V. Uspenskii.

111.1.13. Proposition. Every space X with a countable network can be homeomorphically embedded in a space Cp(Y), with Y some (depending on X) metrizable space with a countable base.

Proof. The space CC(X) has a countable network (chapt. I). Thus [661, there is a metrizable space Y with a countable base which can be continuously mapped onto Cp(X ). Then the space CDCp(X) is homeomorphic to some subspace of Cp(Y) (chapt. 0). In turn, X is homeomorphic to a subspace of C9C9(X). Hence X can be homeomorphically embedded in CC(Y). In particular, every countable space can be realized as a subspace of a space CC(X), with X a space with a countable base. Theorem 1.11 allows us to obtain in an easy manner the following useful result (see the examples 1.7 and 1.8).

111.1.14. Theorem. Every space X with a countable base is an Eberlein-Grothendieck space:

Proof. We may assume that X is a compactum with a countable base. By theorem I.1.1, the space Cp(X) is separable. Fix a countable everywhere dense subspace Y in Cp(X). Then Y separates the points of X, and under the canonical map 0: X --> CC(Y) the compactum X is homeomorphically mapped onto its image ?b(X) C C9(Y). However, Y is o-compact, and by theorem 1.11 there is a compactum 4i such that Cp(Y) is homeomorphic to a subspace of C,,(4?). Hence Ii(X) and X are

Eberlein-Grothendieck spaces. Theorem 1.14 will be considerably strengthened in §4.

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97

2. Okunev's theorem on the preservation of o-compactness under t-equivalence At the end of §1 we have established that if a space Y is a-compact, then there is a compactum X such that CC(Y) is homeomorphic to a subspace of Cp(X ). 0. C. Okunev has proved the converse theorem. Below we will give his reasoning. For a space X we denote by IC(X) the smallest class of spaces containing X and all compacta, and closed under taking finite products, free unions of countable families, transition to closed subspaces, and transition to continuous images.

111.2.1. Theorem (0. G. Okunev). If C,(Y) is homeomorphic to a subspace of a space Cp(X), then Y E JC(X;l. We will prove a somewhat more general result.

In this section we denote by I not the interval 10, 11, as we usually do, but the interval (-1,11. The function on X that is identically zero is denoted by Ox. A set F of realvalued continuous functions on a space X is called D-separating if

Ox E F, f (X) c [-1, 11 = I for all f E F, and the following condition is fulfilled: whatever the closed set P in X, finite set {x1.... of points in X disjoint from P, and number e > 0, there is a function f E F such that if (xi) I < e, for i = 1'... , n, and If (x) I E [3/4,1) for all x E P. Clearly, every D-separating set of realvalued continuous functions on X generates

the topology on X. Let F be an arbitrary set of continuous maps from X into the interval I = 1-1, 1). We denote by ZF(X) the subspace of the product IF consisting of all functions 0: F --+ I such that 4)(Ox) = 0 and there is a neighborhood VV of Ox in F for which 4)(V) C 1-1/2,1/21.

111.2.2. Proposition.

If F is a D-separating set of maps from X into I, then X

is (canonically) homeomorphic to a closed subspace of ZF(X ).

Proof. Under the canonical map zb: X --+ Cp(F) the space X is homeomorphically mapped onto the subspace 71'(X) of Cp(F), since every D-separating family of functions on X generates the topology on X. We have 7P(X) C IF, since f (X) C 1-1, 11 for all f E F. Clearly, if g = ib(x), then g(OX) = Ox(x) = 0 and g is a continuous function on F. Consequently, g E Zp(X), i.e. z/i(X) C ZF(X ). We show that tp(X) is closed in ZF(X). Let ¢ E ZF(X) \ O(X). By the definition of ZF (X ), there are x1i... , x,, E X and an e > 0 such that if f E F and If (x;) l < e for i = 1,...,n, then W5(f)l < 1/2. Since q5(x;) # 2b(x;) for i = 1,...,n and the map 0: X CC(F) is continuous, we can choose an open set U in X such that 0 0 ?i(U). We now fix a function g E F for which Ig(x;)[ < e for i = 1, ... , n and g(X \ U) C [3/4,1. Then 1¢(g)l < 1/2 and 3/4 < G(x)(g)I < 1 for all x E X \ U. Consequently, 0 0 4(X \ U). We conclude that 0 0 7)(X), i.e. tp(X) is closed in ZF(X).

111. 2.3. Proposition. Let F and 4) be families of continuous functions on spaces X and Y with values in 1-1,11, respectively. Moreover, suppose 4; can be homeomorphi-

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rally mapped onto a subspace of F such that the function Oy is mapped to 0x. Then the space ZF(X) can be continuously mapped onto the space Z4,(Y). This assertion follows from the definitions of ZF(X) and Z,(Y) and the following lemma.

111.2.4. Lemma. Let S be a subspace of a space T, and let a point p E S be given. By.F(T) (respectively, F(S)) we denote the set of all functions f on T (S) with values in [-1,11 and satisfying the conditions: f (p) = 0, and there exists a neighborhood U of p in T (S) for which f (U) C [-1/2,1/21. Then the set .F(T) is mapped onto the set .F(S) under the restriction map, and this map is continuous if F(T) and F(S) are endowed with the topologies of pointwise convergence.

Proof. It is clear that .F(T) is mapped into .F(S) by restriction. We only have to verify that each function g E .F(S) is the restriction of a function f E F(T). Take an (open) neighborhood V of p in S for which g(V) C 1-1/2,1/21. There is an open set U in T such that U fl S = V. For all x E S we put f (x) = g(x). If x E T\ S, then we put f (x) = 0. Clearly, f IS = g and f (T) C [-1,11, f (U) C [-1/2,1/21, i.e. f E .F(T).

III.2.5. Proposition. For any family F of continuous functions on X with values in I the space ZF(X) belongs to the class 1C(X).

B(xl,... , xn) the set of all functions 0: F --. I such that if f E F and I f (xl )I < 1/n,... , If (xn)I < 1/n, Proof. F o r a n arbitrary finite collection x 1 , . . . , x we denote by

then (¢(f)t < 1/2. Clearly, B(xl,...,xn) C ZF(X) C IF. For n E N+ we put

Bn = U{B(xl,... ,xn): x1, ... , x,. E X}. The definitions of ZF(X) and the topology of pointwise convergence imply that ZF(X) = U{Bn: n E N+}. Consequently, the space ZF(X) is a continuous image of the free union of its subspaces Bn, n E N+. Since the class K(X) is closed under the operations of taking a free union and transition to a continuous image, it suffices to verify that every Bn belongs to 1C(X).

Consider for n E N+ the set P. = {(xI, ... , xn, 0) E Xn X IF: 0 E B(xl,... , xn), (XI, .. , xn) E Xn}. The definition of Bn makes it clear that Bn is the image of Pn under projection of the product Xn X IF onto the second factor. We verify that P. is closed in Xn x IF. Let (a,... , an, ¢) E (Xn X IF) \ Pn. Then 0 0 B(al,... , an), -

i.e. there is a function fo E F for which I foal )I < 1/n,... , (fo(an)l < 1/n, but 10(fo)l > 1/2. Since fo is continuous on X and IF is endowed with the topology of pointwise convergence on F, the set W = {(b1,... , bn, ¢') E Xn X IF: I fo(bi)! < 1/n, i < n, l '(fo) > 1/2} is open in X n x I F. Clearly, W contains the point (a1,.. . , an, 4); W does not intersect with Pn (by the definition of Pn). Consequently, P is closed in Xn X IF But Xn X I' E 1C(X), since X E 1C(X) and I' is compact. Thus Pn E ?C(X). But then Bn E 1C(X) as the continuous image of Pn (under projection onto IF). It is now easy to prove

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99

111.2.6. Theorem. If some D-separating subspace 4) C C9(Y) is homeomorphic to a subspace of CC(X), then Y E K(X).

Proof. If F can be homeomorphically embedded in Cp(X), then, since C,,(X) is topologically homogeneous, we can take a homeomorphism from F into Cc(X) for which Oy is mapped to Ox. However, C'(X) can be homeomorphically mapped into CC(X, I) in such a way that the point Ox is mapped to itself. Hence for proving theorem 2.6 it suffices to show the following assertion: a) If 4) is a D-separating subspace of C,,(Y) and 4) can be homeomorphically mapped onto a subspace F of Cp(X,I) in such a way that Oy is sent to Ox, then Y E K(X). We prove a). Put Z = C9(X, I). By proposition 2.5, ZF(X) E 1C(X). By proposition 2.3, the space ZF(X) can be continuously mapped onto the space Zb(Y). Thus Z,.(Y) E K(X). But by proposition 2.2, the space Y is a closed subspace of the space 7,4,(Y). Hence Y E K(X). Theorem 2.6 has been proved. Theorem 2.1 is an obvious consequence of theorem 2.6. The subspace Cc(YIX, 1) of C,(Y), consisting of all continuous functions on Y with values in I which can be extended to realvalued continuous functions on X, is an important example of a D-separating set of functions on Y. Theorem 2.6 implies

111.2.7. Theorem. Let YI be a subspace of a space X,, with Cc(Y, JXi) homeomorphic to a subspace of a space Cp(X). Then Y, E IC(X).

Theorem 2.7 allows us to obtain a number of results on the preservation of topological properties under t-equivalence. To this end we apply the well-known

111.2.8. Proposition.

a. If a space X is a-compact, then all spaces in the class K(X) are a-compact. b. If X is a Lindelof E-space, then all spaces in the class K(X) are Lindelof E-spaces. c. If X is a IC-analytic space, then all spaces in the class IC(X ) are IC-analytic.

Recall that a space is called K-analytic if it is a continuous image of a space of type K 5, i.e. of a space that can be represented as the intersection (in some ambient space) of a countable family of c*-compact spaces [3[. Using proposition 2.8,we obtain from theorem 2.7 the following consequences.

111.2.9. Corollary. Let Y, be a subspace of a space X1, with Cp(Y1 1 X1) homeomorphic to a subspace of CC(X). Now: a) if X is then Yi is a-compact; b) if X is a Lindelof E-space, then Y1 is a Lindelof E-space; c) if X is a IC-analytic space, then Y, is_)C-analytic.

Continuous images of the space of irrational numbers are called analytic spaces. It is well known that a K-analytic space with a countable network is an analytic space, and that every analytic space is K-analytic. Since the property that, a space has a

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countable network is preserved under t-equivalence (see chapt. I, §1), assertion 2.9b) implies

111.2.10. Corollary.

Let Y7 be a subspace of a space XI, and Cp(Y1IXI) homeomorphic to a subspace of CC(X), where X is an analytic space. Then Y7 is an analytic space.

Assertions 2.9 and 2.10 imply in specific cases:

III.2.11. Corollary. Let Y1 be a subspace of a space XI, Y2 a subspace of a space X2, with Cp(Y1IXI) homeomorphic to Cp(Y2IX2). We have: a) if YI is a-compact, then Y2 is a-compact; b) if Y1 is a Lindelof E-space, then Y2 is a Lindelof E-space; c) if Y1 is a )C-analytic space, then Y2 is 1C-analytic; d) if Y1 is an analytic space, then Y2 is an analytic space.

111.2.12. Corollary (0. G. Okunev). The relation of t-cyuivolcncc ptrserves acompactness, analyticity, !C-analyticity, and the property of being a Lindelof E-space.

111.2.13. Problem. Let X `' Y (i.e. CDCp(X) is homeomorphic to CpCp(Y)), and X a a-compact space. Is then Y a-compact? The similar questions for Lindelof E-spaces and for analytic spaces are open.

We know that the cardinality of a space is preserved under t-equivalence (see chapt. I, §1). However, it is not known whether it is preserved under t2-equivalence. Corollary 2.12 allows us to obtain the following result.

If X t' Y (i.e. CpC,,CC(X) is homeomorphic to CCCC(Y)), and X is a finite set, then Y is a finite set.

111.2.14. Theorem (A. V. Arkhangel'skii).

Proof. If X is finite, then T = Cp(X) is a locally compact space with a countable base. The space CS(T) of all realvalued continuous functions on T in the topology of uniform convergence on compacts is separable and metrizable by a complete metric. Hence CS(T) is a continuous image of the space of irrational numbers, i.e. CS(T) is analytic. But then so is the space CC(T) = CCC(X), being a continuous image of CS(T). It follows from X 1 Y that CCC(X) ... CCC(Y). Hence, by corollary 2.12, the space CPCp(Y) is analytic. J. Calbrix has proved the following theorem: if the space CC(Z) is analytic, then Z is a-compact. Hence Cp(Y) is a-compact. But then (see chapt. 1, §1) Y is finite. The following assertion is obvious, and, in essence, well known.

111.2.15. Proposition.

On every space Z of weight < r there is a D-separating family of continuous functions of cardinality < -r. Proposition 2.15 and theorem 2.6 imply the following results.

2. OKUNEV'S THEOREM

101

If a space Y is not a-compact, and if the weight of Y is r, then Cp(Y) contains a subspace of cardinality < r which is not homeomorphic to a subspace of a space Cp(X) for any v-compact space X.

111.2.16. Corollary.

111.2.17. Corollary. If a space Y is not o--compact but has a countable base, then Cp(Y) contains a countable subspace which is not an Eberlein-Crothendieck space.

111.2.18. Corollary. If a space Y is not a Lindelof E-space, and if the weight of Y is r, then Cp(Y) contains a subspace of cardinality < r which is not homeomorphic to a subspace of a space Cp(X) for any Lindelof E-space X. Similar assertions hold for analytic and IC-analytic spaces. If Y is a zero-dimensional space, i.e. has a base of open-closed sets, then the set Cc(Y, D) of all continuous maps from Y into the discrete colon V = {0,1 } is a separating family of functions on Y. Hence theorem 2.6 implies

111. 2.19. Corollary. Let Y be a zero-dimensional space, with Cp(Y, V) homcom.orphic to a subspace of a space CC(X). If X has one of the properties: o-compactness, analyticity, IC-analyticity, be a Lindelof E-space, then Y has the same property. We give yet another consequence of theorem 2.6, which is due to O. G. Okunev.

111.2.20. Corollary. If two spaces X and Y are t-equivalent, and X is o-bounded (i.e. is the union of a countable family of bounded-in-itself spaces), then Y is obounded.

Proof. Since X ,., Y, we have v X £ vY (see chapt. If, §7), i.e. the HewittNachbin realcompactifications of X and Y are t-equivalent. It now remains to refer to corollary 2.12 and

111.2.21. Proposition. A space X is o-bounded if and only if its Hewitt-Nachbin realcompactification vX is v-compact.

Proof. Let F C vX, and F bounded in vX. Then F fl X is bounded in X. In fact, every realvalued continuous function f on X can be extended to a realvalued continuous function f on v X. The set i(F) is bounded in R, hence so is the set f (F fl X) = f (FIX ). Therefore, if v X = U{Fi: i E N+}, with all F, compact, then X = U{Fi fl X: i E N+}, with all F,1 fl X bounded in X, hence X is o-bounded. Conversely, let X be o-bounded, i.e. X = U{Xi: i E N+}, with all Xi bounded in X. Then every Xi is also bounded in vX. Thus, the closure F1 = Xi of Xi in vX is compact (see, e.g., [121, [661). We show that vX = U{Fi: i E N+}. Clearly, XCU{Fi: iEN+}. Therefore, if W=vX\U{F: iEN+} 0, then W is a nonempty set of type Ge in vX which is disjoint from X, a contradiction with a fundamental property of the realcompactification vX. Thus, W = 0, i.e. vX = U{Fi:

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i E N+} is a o-compact space. Proposition 111.2.21 has been proved. 111.2.22.

Problem (0. G. Okunev).

Let X L Y and X o-pseudocompact

(o-countably compact). Is then Y o-pseudocompact (respectively, o-countably compact)? The following assertion is rather useful.

111.2.23. Corollary (0. G. Okunev).

The following assertions about a space

CC(Y) are equivalent: a) Cp(Y) is homeomorphic to some topological subspace of the space Cp(J), where

J is the space of irrational numbers; h) Cp(J) contains a linear topological subspace (not necessarily closed) which is linearly homeomorphic to Cc(Y).

Proof. If Cp(Y) is homeomorphic to a subspace of Cp(.I), then Y is analytic, by corollary 2.10. But then J can be continuously mapped onto Y, hence (see chapt. 0) the space Cp(Y) is linearly homeomorphic to a linear topological subspace of CC(J). We denote by e'(X) the supremum of the cardinalities of closed discrete subspaces of the spaces X", where n E N+. The definition of the class K(X) readily implies

111.2.24. Proposition. If e'(X) < -r, then e'(Y) < T for all Y E K(X). Combining proposition 2.25 and theorem 2.6, we arrive at the following results of O. G. Okunev:

111.2.25. Corollary.

If Cp(Y) is homeomorphic to a subspace of a space Cp(X),

then e* (Y) < e* (X).

111.2.26. Corollary. If X t Y, then e* (X) = e* (Y).

3. Compact sets of functions in Cp(X). Their simplest topological properties Every compactum F can be represented as a subspace of some Cp(X): it suffices to take a discrete space X of cardinality r, with T the weight of F. It is quite natural to pose the question: how can the compact subspaces of Cp(X ), with X a compactum, be characterized? In this section we give the simplest properties of such compacta. A compactum F is called an Eberlein compactum if there is a compactum X such that F is homeomorphic to a subspace of Cp(X). Thus, the Eberlein compacta are compacta that are Eberlein-Grothendieck spaces (see §1). Later we will give a number of equivalent definitions of Eberlein compacta. A particular case of theorem 1.13 is

3. COMPACT SETS OF FUNCTIONS IN Cp(X). THEIR SIMPLEST TOPOLOGICAL PROPERTIES 103

111.3.1. Theorem. Every metrizable compactum is an Eberlein compactum. However, the class of Eberlein compacta is substantially larger than the class of metrizable compacta. To exhibit examples of nonmetrizable Eberlein compacta we use a simple construction, which is described in the proof of the proposition below. Recall that AT denotes the one-point compactification (in the sense of P. S. Aleksandrov [661) of the discrete space of cardinality r > lto. For all r > Ko the space

A, is a Frechet-Urysohn compactum with unique isolated point a,.. For r > Eo the compactum A7 is not metrizable [661.

111.3.2. Proposition.

If a space X contains a disjoint family {Up: a E Al} of nonempty open sets such that IMI = r > No, then Cp(X) contains a subspace which is homeomorphic to the compactum Ar.

Proof. For each ( E A'l we choose f E C,,(X) such that. I E fn(U,,) and {0}. We denote the function on X which is identically equal to 0 by g. 'T'hen f

a E Al} is a discrete (in itself) subspace of Cc(X), and the subspace F = if,,: a E All U {g} C CP(X) is compact, which can be readily verified. Since IMO = 7, we conclude that. F is homeomorphic to AT. In particular, the assumptions of proposition 3.2 are fulfilled if X is taken to be A,., the disjoint family {U0: a E All is constituted by the open singleton subsets. Hence AT is homeomorphic to a subspace of CO(AT). Since AT is a compactum we obtain

111.3.3. Proposition 170]. For every r, A, is an Eberlein compactum. The following theorem allows its to further enlarge our insight in the class of Eberlein compacta.

111.3.4. Theorem.

The product of a countable family of Eberlein compacta is an

Eberlein compactum.

Proof. Let F,, C C,,(X,,), where F. and X,, are compacta, n E N+. The product F = fj{F,,: n E N+} is homeomorphic to a subspace of the space C,(Y), where Y = F-®{X,,: n E N+} is the free sum of the spaces X (since C'(Y) is homeomorphic Hence (see to the product jj{CP(X, ): n E N+}, see chapt. 0). But Y is §2), there is a compactum Z such that Cp(Y) (and thus F) is homeomorphic to a

subspace of CP(Z). We note the obvious 111.3.5. Proposition. Eberlein compactum.

Every closed subspace of an Eberlein compactum is an

We will later prove that a continuous image of an Eberlein compactum is also an

104

fit. TOPOLOGICAL PROPERTIES OF FUNCTION SPACES

Eberlein compactum. The proof of this result, which belongs to S. P. Gul'ko and M. E. Rudin, is highly nontrivial. We see that the Eberlein compacta form an extension of the class of metrizable compacta that is very satisfactorily from the categorical point of view. We now establish the simplest topological properties of Eberlein compacta.

111.3.6. Theorem.

Every Eberlein compactum is a monolithic Frechet-Urysohn space and satisfies the first axiom of countability on an everywhere dense set of points. Proof. Let F be an Eberlein compactum, i.e. F is a compactum and F C Cp(X ), with X a compactum. Then t(F) < t(CC(X)) < 1 o, and Cp(X) is monolithic since X is stable (see chapt. 1, §4, and chapt. II, §6). Consequently, F is a monolithic compactum of countable tightness. It remains to prove that every monolithic compactum of countable tightness satisfies the conclusions of theorem 3.6.

111.3.7. Proposition.

Every monolithic compactum F of countable tightness is a

Frechet-Urysohn space.

Proof. Let y E F, A C F, and y E A. There is a countable set B C A such that y E B. Since F is monolithic, B is a space with a countable network. But B is compact, hence (see [66]) is a space with a countable base. Thus there is a sequence of points in B C A converging to y. 111.3.8. Proposition [3]. Every monolithic compactum of countable tightness satisfies the first axiom of countability on an everywhere dense set of points.

In the proof of proposition 3.8 a key role is played by the following lemma.

111.3.9. Lemma [3].

Let X be a compactum of countable tightness and U a nonempty open set in X. There are a nonempty closed set P of type G6 in X and a countable set A C X such that P C A fl U. Proof. Since X is a regular space, we may assume that U = X. We assume that the key lemma is not true, and give a construction by transfinite recursion which leads to a contradiction. Choose an arbitrary point xo E X and put F0 = X. Let Q < w(, and suppose that for all a < Q a point x0 E X and a nonempty closed set F. of type G6 in X have been chosen, such that, moreover, F0. C F0.. if a" < a'. Consider the sets Ap = {x0: a < 6} and 4>p = fl{F0: a < ,6}. Since X is a compactum and {F,,,: a < ,Q} is a chain of nonempty closed sets in X, the set 4ip is not closed. Clearly, d>p is a closed set of type G6 in X, while A0 is countable. By definition 4>p \A# is nonempty. Hence there is a nonempty closed set Fp of type G6 in X such that Fp C (Dp \ A0. Choose xp E Fp arbitrarily. This finishes the construction of the transfinite sequences {x0:

a < w1} and {F0: a < wl}. It is obvious from this construction that the following

3. COMPACT SETS OF FUNCTIONS IN Cp(X). THEIR SIMPLEST TOPOLOGICAL PROPERTIES 105

conditions are satisfied:

1) x, EFafor all a<wl; 2) Fpn{xa: a
4) {xa: a>/3} C Fpfor all 6<w1. Properties 2) and 4) imply:

5) {xa: a<,0}n{xa: a:?/3}}=0 forall6<w1. (Property 5) means that {xa: a < wi} is a free sequence in X in the sense of [16].)

By 5), xa- 0 xa,, if a' # a"; consequently, the cardinality of the set A = {xa: a < wi} is fti. Since X is compact, there is in X a complete accumulation point x' for A. Then x.' E A, and t(X) 1 o implies that x.' E {xa : a < /3'} for some /3' < w1. Put V == X \ {xa : a > /3'). Property 5) implies that V is a neighborhood of x' which intersects A in the countable set {xa: a < )3'1, contradicting the fact that x.' is a complete accumulation point for the uncountable set A. Lemma 3.9 has been proved.

Proof of proposition 3.8. Let U be a nonempty open set in a monolithic compactum

X of countable tightness. By lemma 3.9 there are a countable set A C X and a nonempty closed set F of type G6 in X such that F C X n U. But A is a compactum with a countable base, since every monolithic compactum is strongly monolithic [141. Consequently, F is also a compactum with a countable base. Moreover, every point x E F is of type G6 in F, hence also in X. But every point of type G6 in a compactum has a countable base in this compactum [16], [66]. Thus X satisfies the first axiom of countability at all points of the set F C U. Proposition 3.8 and theorem 3.6 have been proved.

Recall that a space X is called topologically homogeneous if for any two points x, y E X there is a homeomorphism f of X onto itself such that f (x) = y. Clearly, if a topologically homogeneous compactum satisfies the first axiom of countability at at least one point, then it satisfies the first axiom of countability at all points. Hence theorem 3.6 implies

111.3.10. Corollary. Every topologically homogeneous Eberlein compactum satisfies the first axiom of countability at all points.

The cardinality of every compactum satisfying the first axiom of countability does not exceed 2x0. We obtain 111.3.11. Corollary [16]. The cardinality of every topologically homogeneous Eberlein compactum does not exceed 210.

The example of the space A? makes it clear that there are Eberlein compacta of arbitrarily large cardinality. An example of a nonmetrizable Eberlein compactum

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satisfying the first axiom of countability at all points is provided by the 'two-circle' compactum of P. S. Aleksandrov [16]. Jan van Mill has constructed a topologically homogeneous nonmetrizable Eberlein compactum (120). A useful result has been obtained by M. E. Rudin: she has proved that if a compactum is the union of two metrizable subspaces of it, then it is an Eberlein compactum. The one-point compactification of the Isbell-Mrowka space ' [66] is an example of a compactum that is representable as the union of three metrizable subspaces but is not an Eberlein compactum. This compactum is not monolithic, and is not a Frechet-Urysohn space. Under the assumption that 2"0 < 21`11, every Frechet-Urysohn compactum satisfies the first axiom of countability at the points of an everywhere dense set [3]. However, as has been proved by V. 1. Malykhin, the system ZFC of axioms of set theory does not contradict the existence of a nonempty Frechet-Urysohn compactum which does not satisfy the first axiom of countability at any point. These remarks make it possible to better judge the role of monolithicity in assertions 3.6 and 3.7. In general topology, another extension of the class of met.rizable compact.a is very popular, to wit, the class of dyadic compact [16], [66]. It is well known that every dyadic compactum of countable tightness is metrizable [66]. This and theorem 3.6 imply

111.3.12. Theorem. Every dyadic Eberlein compactum is m.etrizable.

4. Grothendieck's theorem and its generalizations Recall (chapt. I, §1) that a set A C X is called bounded in a space X if every realvalued continuous function on X is bounded on A. Clearly, if A is bounded in X, then its closure is also bounded. A pseudocompact space is bounded in any ambient space, and boundedness of a space in itself is equivalent to pseudocompactness. On the other hand, if X is pseudocompact, then every set A C X is bounded in X. Since an arbitrary space can be embedded as a closed subspace in some pseudocompact space, we conclude that every space X can be realized as a closed bounded set in some other space. From this it is clear how far away the relative notion of boundedness is from its absolute version pseudocompactness. The following successful generalization of a well-known theorem of Crothendieck is due to M. O. Asanov and N. V. Velichko [20].

III.4.1. Theorem. If a space X is countably compact, then the closure F in CC(X) of any set F bounded in CC(X) is a compactum in CC(X). This theorem has numerous consequences, and takes a key place in many results in this and the following chapter. In particular, it immediately implies

111.4.2. Corollary.

If X is countably compact, then every closed pseudocompact subspace of CP(X) is compact.

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111.4.3. Corollary. Let X be countably compact, and let A C CC(X) be a countably compact set in CC(X) (i.e. whatever the infinite set B C A, the space CC(X) contains a limit point of B). Then the closure of A in CC(X) is compact.

This is the classical statement of Grothendieck's theorem. The proof of theorem 4.1 is preceded by some general assertions, among which the first and the third are obvious.

111.4.4. Proposition.

If X is a normal space, then every closed bounded subspace

of X is pseudocompact.

For paracompact spaces pseudocompactness is equivalent to compactness. Hence we have

111.4.5. Proposition. In a paracompact space every closed bounded set is comnpmct. 111.4.6. Proposition.

Under a continuous map f : X - Y the image of a bounded set in X is a bounded set in Y.

111.4.7. Lemma. Let X be a space, F a bounded set in Cp(X), and Y a countable subspace of X. Let 4 y denote the closure of the set Fy = { fly: f E F} in Cp(Y). Then 4y is compact, and hence closed in the space W' D Cp(Y). Proof. The restriction map 7r: C P(X) - Cp(Y) (under which an f E Cp(X) is sent to its restriction f it to the space Y) is continuous, and 7r(F) = Fy. By proposition 4.6 the set Fy is bounded in Cp(Y). But Cp(Y) C BY, and B is a space with a countable base since Y is countable. Thus Cp(Y) is paracompact, and by proposition 4.5 44y is compact.

Proof of theorem 4.1. The set F is pointwise bounded, i.e. for every x E F the set f f (x): f e F) is bounded in R. In fact, otherwise for some x the continuous function V1=: Cp(X) --1 R defined by V' (f) = f(x) would be unbounded on F. Put Br, = If (x) : f E F} for x E X. Then Bx is compact, and F c fl(B?: x E X} C Rr,

where fl{B=: x E X} is the Tikhonov product of the compacta B, and is thus compact itself. Thus, the closure of F in Rx is a compactum, P. We show that. P C Cp(X), which will finish the proof of theorem 4.1.

Assume that P \ Cp(X) # 0. and fix an f E P \ Cp(X ). Then f : X --a R is a discontinuous map, and hence there are a point x' E X and a set A C X such that x' E A but, f (x) f (A). `lake open sets U and C in R for which f (x') E U,

f(A)cC,and 1nZ7 0.

We determine a sequence {xn: n E N+} of points in A, a sequence {Vn: n E N+} of open sets in X, and a sequence f f,,: n E N+} of elements of F such that for all n E N+: 0) x' E V n ; 1) Vn+1 C V,,; 2) f2(Vn) C Ui 3) fnti(xi) E C f o r i = 1, ... , n.;

and 4) x, Vn.

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Since f belongs to the closure of F and f (x') E U, there is an f, E F such that f&*) E U. Put Vi = fi I (U). Then VI E) x' and VI is open. Consequently we can choose x1 E VI fl A. There is an f2 E F such that f2(x') E U and f2(x1) E G. Take V2 to be an open set in X such that x' E V2 C f2 I(U) and V2 C VI, and take x2 E V2 fl A. By continuing the constructions in the obvious way for n E N+, we arrive at the required sequences. Since X is countably compact, there is a point x,, E X which is a limit point of the sequence {xn: n E N+}. Moreover, x E 4) = fl{Vn: n E N+}. In fact, by 2), 4i = fl{Vn: n E N+}. Properties 4) and 1) imply that xi E Vn for all i > n. Thus and x,,. E 4). We have E fn(fl{Vn: n E N+}) C fn(Vn) C U. Put Y = U {xn: n E N+} and gn = fnly for all n E N+. Then the closure of the set {gn: n E N+} in CC(Y) is compact, by lemma 4.7. Thus there is a limit function g E CC(Y) for the sequence {gn: n E N+}. Since gn(x;) = fn(x;) E G for

i>n(by4)and 1)),we have g(X;)EGfor all iEN+. E 11 implies E C.. On the. other hand, g E U. We obtain E U fl = 0, a contradiction. This theorem allows substantial generalizations. In particular, such generalizations can be found in the already mentioned work [20] of M. O. Asanov and N. V. Velichko, and also in the articles [1], [139], [82], [137]. Here we give generalizations of Grothendieck's theorem in three directions. A space X is called countably pracompact if there is a subspace Y C X which is everywhere dense in X and countably compact in X in the following sense: every infinite set A C Y has a limit point in X. Clearly, every countably pracompact space is pseudocompact, but the converse does not hold.

Consequently

For the first generalization of Grothendieck's theorem we need the following obvious lemma.

111.4.8. Lemma.

Let X, Z be topological spaces, f : X -- Z a map, Y C X,

Y everywhere dense in X, and let the restriction of f to every subspace of the form Y U {x}, where x is an arbitrary point in X, be continuous. Then the map f itself is continuous.

Lemma 4.8 allows us to choose in the proof of theorem 4.1 the set A to lie within a given everywhere dense subspace Y of X. This makes it clear that theorem 4.1, while preserving all simple details of its proof, can be generalized to the following theorem.

111.4.9. Theorem.

If a space X is countably pracompact, then every bounded set in CC(X) has compact closure.

Recall that in chapt. I, §2 we have given an example of a pseudocompact space X such that CC(X) contains a closed pseudocompact (moreover, bounded) subspace which is not compact. Thus, theorem 4.1 cannot be generalized to pseudocompact

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109

spaces.

For further generalizations of Grothendieck's theorem it is convenient to introduce the following notions.

Let M be a family of subspaces of a space X. We say that X is functionally generated by the family M if the following condition holds: for every discontinuous function f : X --+ R there is an A E M such that the function f IA (the restriction of f to A) cannot be extended to e. realvalued continuous function on all of X. E.g., a space X is functionally generated by the family of its countable subspaces if and only if its R-tightness is countable: tR(X) < fto. This follows directly from the definition of the R-tightness tR given in chapt. II, §4. In the sequel we understand by a placement property (also called position property) any relative topological property of a subspace Y of a space X with respect to the whole space X. A placement property P is called a continuously invariant property if the facts that Y is P-placed in X and f : X --' Z is a continuous map imply that f (Y) is P-placed in f (X). E.g., closedness of Y in X is riot a continuously invariant property, while houndedness of Y in X is a continuously invariant property by proposition 4.6. These notions allow us to formulate a very general principle, from which generalizations of theorem 4.1 in various directions can be easily obtained. If Y is a subspace of a space X, then lry will now denote both the restriction map

R" -+ Ry and the restriction map CP(X) --+ Cp(Y), defined by iry(f) = f ly. If A C CP(X), then AlY denotes the subset Try = If iy: f E Al of the space CP(YI X ).

111.4.10. Proposition. Let a space X be functionally generated by a family M of subspaces of it, let P be some continuously invariant placement property, and let the following condition be fulfilled:

a) if Y E M, then the closure B in CP(YIX) of every P-placed set B E CP(YIX) is compact.

Then the closure in CP(X) of every P-placed set A E CP(X) coincides with the closure of A in Rx, and is compact. Proof. Take an arbitrary P-placed set A E CP(X). Let f : X --+ R he a discontinuous function, and f E A (in Rx). We deduce from this assumption a contradiction. There is a Y E M such that the function fir cannot be extended to a continuous

function on all of X. Then fly 0 CP(YIX). Since the map lry: Rc -+ Ry is continuous and f E A, we have fly E AMY (in R'). Property P is a continuously invariant property, therefore the set AIY = iry(A) is P-placed in CP(YlX). By condition a), the closure 4) of AMY in CP(YIX) is compact, hence the closure of 4i in Ry coincides with 4 and belongs to CP(YI X). But fir E AIY C 4' (in Ry). Thus, f 11, E CP(YIX), contradicting the choice of f and Y. The last part of the conclusion of proposition 4.10 is a consequence of the pointwise boundedness of the family A, i.e. of the fact that the set { f (x): x E Al is bounded in R for every x E X.

(This follows since UM = X, and hence x E Y for some Y E M: if the set If (x):

110

III. TOPOLOGICAL PROPERTIES OF FUNCTION SPACES

f E A} would be unbounded, then {fly : f E Al would not be compact.) The applications of proposition 4.10 are related to the following notion. A placement property P is called a property of boundedness type if it is a continuously invariant property and the following condition is satisfied: /3) in every space with a countable base the closure of every P-placed set is compact.

111.4.11. Proposition.

A continuously invariant placement property P is a property of boundedness type if and only if it implies boundedness, i.e. every P-placed subspace Y in X is bounded in X.

Proof. In fact, condition /3) is satisfied for every bounded subset of a space with a countable base. Conversely, let Y C X, Y P-placed in X, and P a property of boundedness type. Let f be a realvalued continuous function on X. Then f (Y) is P-placed in f (X), and by condition /3) the closure of f (Y) in f (X) is compact. Consequently, f (Y) is it hounded set. in R., i.e. (lie function f iS hoinided on Y, and Y is bounded in X. We list (and at the same time denote) some properties of boundedness type which are related to our further exposition. Pa: every realvalued continuous function on X is bounded on Y (boundedness);

PP: the closure of a set Y in X is a pseudocompact space (pseudocompact placement); P.«: every infinite set A C Y has a limit point in X (relative countable compactness); Pcco.: the closure of a set Y in X is a countably compact space (countably compact placement); Pcom: the closure of a set Y in X is compact (compact placement); Pc .u: the closure of a set Y in X is a Fr4chet-Urysohn compactum. The following two modifications of theorem 4.1 follow from proposition 4.10 in a most simple manner.

111.4.12. Theorem. Let the R-tightness of a space X be countable, i.e. the family of all countable subspaces of X functionally generates X. Then every bounded set in CP(X) has compact closure in CP(X). Proof. Take for .M the family of all countable subspaces of X. Then every CP(YIX), where Y E M, has a countable base, and hence every bounded set in CP(YIX) has compact closure. It remains to apply proposition 4.10. We may also reason quite differently. As has been proved in chaps. II, §4, countability of the R-tightness of X implies that CP(X) is realcomplete. But then every bounded dosed set in CP(X) is compact.

111.4.13. Theorem. Let X be a space which is functionally generated by the family of its compact subspaces. Then every bounded set in CP(X) has compact closure in

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III

llp(X). Proof. We take for M the family of all compact subspaces Y of X and use the fact that. Cc(YIX) = Cp(Y), since every realvalued continuous function on a compact subspace can be extended to the whole space. Now the proof follows from proposition 4.10 and theorem 4.1. We say that a family M of subspaces of a space X strongly functionally generates

X if there is for each realvalued discontinuous function f on X a Y E M such that the function f Iy: Y -+ R is discontinuous. Proposition 4.10 has the following modification.

111.4.14. Proposition. Let a space X be strongly functionally generated by some family .M of subspaces Y of X such that every closed hounded set in Cc(Y) is compact. Then every bounded set in C,(X) has compact closure in Cp(X). 'i'he proof of prolunition 4. 10 can be :uutonIRI.ieally mollified to yield a proof or proposition 4.14. Invoking in the proof of theorem 4.13 proposition 4.14 and theorem 4.1, we obtain a proof of

111.4.15. Theorem.

If a space X is strongly functionally generated by the family M of its countably compact subspaces, then the closure of every bounded set in Cp(X ) is compact.

Note that in theorem 4.15 the elements of M are not assumed to be closed in X. All spaces of countable tightness satisfy the assumptions of theorem 4.12; the assumptions of theorem 4.15 are satisfied for all k-spaces, and for all quasi-k-spaces in the sense of Ju. Nagata 1661. Thus, theorems 4.12, 4.13, and 4.15 are generalizations of the corresponding results of M. 0. Asanov and N. V. Velichko 1201 for spaces of countable tightness and quasi-k-spaces. We show that theorem 4.1 cannot be true for a-compact spaces. 111.4.16. Example. In the topological product I"1 = f1{I,,,: a E A}, where JAI = RI and every I,, is the interval [0,1] with the ordinary topology, we consider the subspace Y = QI"1 of all points with only finitely many nonzero coordinates. We fix

a point z E I" with uncountably many nonzero coordinates, and put Z = Y U {z}. The space Y is the union of an increasing sequence of compacts Fn, n E N+. E.g., F. can be taken to consist of all points in al" with at most n nonzero coordinates. As can be readily seen, the space Z has the following properties at z: a) {z} is a set of type G6 in Z; more precisely,

{z}=f1{Z\ Fn: nEN+}; b) the point z is not isolated in Z, i.e. z E Z \ {z}; c) if M C Z, M countable, and z V M, then z M.

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Property c) follows from the fact that z has uncountably many nonzero coordinates, while for every point in Z \ {z} all coordinates, except finitely many, are zero. We now need the following assertion.

111.4.17. Proposition.

Let Z be an arbitrary (Tikhonov) topological space containing a nonisolated point z of type Gs which is not l 0-accessible from Z \ {z}, in the sense that z V M for every countable set M C Z \ {z} Then CC(Z) contains a countable closed discrete subspace B which is bounded in CC(Z).

Proof. Take closed sets Fn in Z for which:

d): U{Fn: nEN+}=Z\{z}; e): Fn C Fn+I for n E N+; Fix for each n E N+ a realvalued continuous function fn on Z such that f): fn(Fn) = {1} and fn(z) = 0. Clearly, the sequence {fn: n E N+) converges pointwise (in RZ) to a (discontinuous) function f which is identically 1 on the set Z \ {z} and vanishes at z. Hence {f} U {f.: n E N+} is a compact subspace of RZ. Since f does not belong to the subspace CC(Z) of RZ, we conclude that B = f fn: n E N+} is an infinite closed discrete subspace of CC(Z). We show that B is bounded in CC(Z). Consider an arbitrary realvalued continuous function g on CC(Z). Take an appropriate countable subspace Y C Z and a continuous function h defined on the subspace Cp(YI Z) _ {fly: f E Cp(Z)} of CC(Y) such that

g(f) = h(flz) for all f E Cp(Z). Without loss of generality we may assume that z E Y. Clearly, the sequence { fnl y: n E N+} converges in Ry to the function fly which is identically 1 on the set Y \ {z} and vanishes at z. The subspace F = { fnl y: n E N+} U {fly} of Ry is thus compact. We show that F C CC(Y,Z). We need only verify that f ly E Cp(YIZ). This we do as follows. The set Y \ {z} is countable. Consequently, by c) the point z is isolated in Y.

Thus, z 0 P, where P = Y \ {z} is a closed set in Z, and there is a function f' E Cp(Z) for which f'(z) = 0 and f'(P) = 1. Clearly, h y = f'ly E Cp(YIZ). The formula g(f) = h(f ly) now implies that g(B) c h(F). Since F is a compactum and h is continuous, we conclude that h(F) is a bounded set in R. Consequently, g(B) is bounded in R, i.e. g is bounded on B. Proposition 4.17 has been proved. Combining proposition 4.17 and example 4.16 gives

111.4.18. Proposition. There is a v-compact space Z such that CC(Z) contains an infinite closed discrete subspace which is bounded in Cp(Z).

Example 4.16 and proposition 4.18 are due to 0. G. Okunev, as is the following assertion.

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There is a o-compact space Z such that the space Cp(Z) is not homeomorphic to a closed subspace of a space Cp(X) for any countably compact space X.

111.4.19. Corollary.

Proof. Let Z be a space as in proposition 4.18, and A an infinite closed discrete subspace of Cp(Z) which is bounded in C,,(Z). Assume that CC(Z) can be represented as a closed subspace of a space Cp(X ), with X countably compact. Then A is bounded and closed in Cp(X), and is an infinite discrete subspace of Cp(X). This contradicts theorem 4.1. It is worthwhile to combine assertion 4.19 with assertion 111.1.11.

There is a nice interaction between theorem 4.1 and the results of the previous section on the structure of Eberlein compacta. We obtain not only interesting consequences, but also further generalizations of theorem 4.1.

111.4.20. Theorem [12).

Let X be a o-compact space. Then every countably compact (in itself) subspace P of C,,(X) is an Eberlein compactum (and is thus closed in CC(X)).

Proof. We may assume (§2) that X is compact. By theorem 4.1, the subspace F = P of CC(X) is compact. Since X is compact, F is an Eberlein compactum. By theorem 3.6, F is a Frechet-Urysohn compactum. This implies that if F0 P, then some sequence = {x,,: n E N+} of points of P converges to a point of F \ P. But then C does not have limit points in P, contradicting the countable compactness of P. We conclude that F = P is compact. It is worthwhile to combine theorem 4.20 with proposition 4.18. Using the above theorems we can establish an unexpected fact, which was first discovered by R. Haydon [104): if X runs through the class of all pseudocompact spaces, then in CC(X) the same compacta arise as in the case when X varies within the class of compacta. More precisely, we have

111.4.21. Theorem. LetX be a a-bounded space, vX its Hewitt-Nachbin realcompactification, and (PT) a countably compact subspace of CC(X). Then the topology T generated on P by Cp(vX) coincides with T; moreover, (P, T) is an Eberlein compactum (as usual, we canonically identify C(X) and C(vX)).

Proof. The identification map 9: Cp(vX) Cp(X) is continuous, hence T C 7. Each infinite set A C P has, by requirement, a limit point a in (P,T). The set A* = A U {a} is countable, so TI and T generate the same topology on A` (see theorem 11.4.29). Consequently, a is a limit point of A also in the space (P, TI). Thus

(P,7) is countably compact. But (P,T1) is a subspace of Cp(vX), and vX is ocompact by proposition 2.21. Theorem 4.20 now implies that (P, T) is compact. But then (P, T) is compact, since T C T1. Theorem 4.21 implies that if a space X is pseudocompact and F is a countably compact subspace of Cp(X), then F is an Eberlein compactum. Indeed, a somewhat

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more refined assertion holds, improving corollary 4.3.

111.4.22. Theorem.

Let X be a pseudocompact space, P C Cp(X), and P countably compact in Cp(X) (see 4.1). Then the closure of P in Cp(X) is an Eberlein compactum.

Proof. As before, we identify the sets C(X) and C(vX). By requirement, each infinite set A C P has in CC(X) a limit point, a. The set A' = A U (a} is countable, hence the topologies of Cp(X) and CC(vX) generate the same topology on A' (see theorem 11.4.29). Consequently, a is a limit point of A also in Cp(vX ), i.e. P is countably

compact in CC(vX). But vX is compact, since X is pseudocompact. Assertion 4.3 now implies that the closure P of P in CC(vX) is an Eberlein compactum. The identification map Cp(vX) --+ Cp(X) is continuous, hence the topologies generated on P by CC(vX) and Cp(X) coincide; in particular, P is the closure of P in Cp(X). The range of applicability of theorem 4.20 can be enlarged as follows.

111.4.23. Theorem.

If a space X contains an everywhere dense a-pseudocompact subspace Y, then every countably compact subspace P of Cp(X) is an Eberlein compactum.

Proof. The space Cp(X) can be condensed onto a subspace of Cp(Y). The image of P under this condensation is a countably compact subspace P' of Cp(Y). By theorem 4.20, P is an Eberlein compactum. Using the fact that Eberlein compacta are Frechet-Urysohn spaces, it remains to refer to the following well-known lemma.

111.4.24. Lemma.

Every condensation from a countably compact space onto a Irechet-Urysohn space is a homeomorphism.

Proof. Let f : X --' Y be such a condensation. Let y E Y, A C Y, and y E A \ A. Since Y is a Frechet-Urysohn space, there is a countable set B C A with y as unique limit point in Y. But then a point x' in X distinct from x = f -'(y) cannot be a limit

point of f''(B). Since X is countably compact, the infinite set f-1(B) does have a limit point in X. Hence x is the limit point of f-'(B). Moreover, x E f-' (A), i.e. the map f-: Y --b X is continuous. Further generalizations of theorem 4.20 have been obtained in [12]. In particular, in [12] it was proved that if a space X contains an everywhere dense a-compact subspace, then every pseudocompact subspace of CC(X) is an Eberlein compactum. In relation to theorem 4.20, the following problem is of interest.

Let Y C X, Y everywhere dense in X, Y a a-bounded 111.4.25. Problem. space. Is it then true that every countably compact subspace of CC(X) is an Eberlein compactum? In relation to the methods used in this section, we note the following result, which

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was obtained in [12). Let X be ,t pseudocompact space (and let C(X) and C(OX) be identified). Then every closed subspace Y of CP(X) containing an everywhere dense in Y and o-compact subspace is (in the same topology) also a subspace of Cp(L3X).

5. Namioka's theorem, and Ptak's approach In this section we establish an unexpected and important fact connected with the relation between the topologies of pointwise and uniform convergence on arbitrary compact sets of functions in CC(X). This fact was discovered by I. Namioka [125[, and a number of very interesting results, essentially involving compactness, is related to it. An important role in the proof of Namioka's theorem is played by a result of V. Ptak, with which we start. To study convergence of functions on compacta, Ptak developed an elementary combinatorial method, allowing us to manage without, invoking deep results from measure and integration theory. We agree on the following notation. If S is a nonempty set, then M(S) is the set of all nonnegative realvalued functions on S satisfying the conditions: 1) the set N(f) = {s E S: f (s) > 0} is finite;

2) E{f(s): 6 ES}=1. These restrictions on f imply that f (s) < 1 for all s E S. For A C S we obtain f (A) = E{ f (s): s E Al.

Let E be a family of subsets of S, and K C S. By S(K) we then denote the set of all W E E for which W f) K # 0. For arbitrary e > 0 and H C 5, we set M(H, E, e) = (f EM(S): N(f) C H and f (W) < e for all W E £}. Using this notation, and without any additional assumptions regarding S and E, the following central lemma holds.

111. 5.1. Lemma. Let T C H C S and 0 < e' < e, with T 0 and M(H, E, e) = 0. Then there is a nonempty finite set K C T such that M(T, £(K), e') = 0.

Proof. Suppose M(T, E(K), e') # 0 for every finite set K C T. Fix a natural number n' such that e' + (1/n*) < e. Take a nonempty finite A, C T, choose an f1 E M(T, £(A1), e'), and put A2 = AIUN(f1). Here, A2 C T. In the (by requirement) nonempty set M(T, £(A2), e') we take an f2, and put A3 = A2 U N{ f2}. Here A3 C T. Performing in this manner n' steps, we obtain a collection Al C C A,,. of finite subsets of T and a collection of functions fl,..., f,,.. We put f = (f, + + f a contradiction. Take an arbitrary W E E. It turns out that all numbers f 1(W ), ... , f,,. (W ), except at most one, do

not exceed e'. In fact, let p be the first index among 1, ... , n' such that fp(W) > 0, and let q > p, q E {1,.. -, n*). Then N(fp) C AP+1 C Ag, w fl N(fp) # 0, hence W fl A. 0 0, i.e. W E E(Aq). Now fq E M(T, £(Aq), e') implies that fg(W) < e'. This

111. TOPOLOGICAL PROPERTIES OF FUNCTION SPACES

116

implies that

f(W) =

n-(1+n`E) =E +4 <e,

i.e. f E M(T, £, e). The proof is finished. Using lemma 5.1 we can prove the following basic

111. 5.2. Theorem. Let S be an infinite set, and E a family of subsets of it. Then the following conditions are equivalent:

1) there are an infinite set H C S and an e > 0 such that M(H, £, e) = 0; 2) there are sequences {sn: n E N+} in S and {W,,: n E N+} in £ such that sn, 7" sari if n'# n", and {SI, ... , Sn} C Wn for all n E N+.

Proof. Let 2) be fulfilled. Consider the set H = {sn: n E N+} C S. By requirement, H is infinite. We show that M(H, £, e) = 0 for 0 < e < 1. Suppose f E M(11,,6, e).

Then N(f) C H, and N(f) C I S ] ,- .. , sn) C 141. E £ for some it E N+. Then I = f (N(f )) < f (Wn) < e < 1, a contradiction. We now assume that M(H, £, e) = 0 for some infinite set H C S and some e > 0. By lemma 5.1 there is a nonempty finite set K1 such that M(H,£(K1),e/2) = 0. Since H is infinite, H \ K1 # 0, and lemma 5.1 implies that there is a nonempty finite set K2 C H \ K1 for which M(H \ K1i £(K1) n £(K2), a/4) = 0. Repeat this reasoning with the nonempty set H \ (K1 U K2), etc. As the result of this simple inductive construction we obtain a sequence of pairwise disjoint nonempty finite sets K1, K2,..., such that Fn = M(H \ (K1 U ... U Kn), £(K1) n n £(K,,), a/2n) = 0 for all n E N+. Then also £(K1) n . . . n £(K.) 34 0, otherwise any function f E M(S) for which N(f) C H \ (K1 U . U K,,) would belong to the set Fn. This allows us to fix for each n E N+ a Un E £ such that Un n K; # 0 for all

i E {1,...,n}. The set K1 is finite. Hence there is an s1 E K1 belonging to infinitely many terms of the sequence CI = {U,,: n E N+}. Let 1;2 denote the subsequence of C1 formed by these terms. Since K2 is finite, some point s2 E K2 belongs to infinitely many terms of the sequence e2. Continuing this reasoning in the obvious way, we obtain the required sequence SI i S2,. - . C S. Now W1 can be taken to be the first term of the sequence 1, W2 the first term of the sequence S2, etc. Theorem 5.2 has been proved. Using theorem 5.2 it is easy to prove an important result related to pointwise and uniform convergence on compacta of sequences of functions.

111.5.3. Theorem.

Let X be a compactum, and C = If,,: n E N+} a sequence of realvalued continuous functions on X such that lim fn(x) = 0 for all x E X, and Ifn(x)I S 1 for all x E X and all n E N+. Then for every a> 0 there are a number p E N+ and nonnegative real numbers a1, ..., Ap such that Ia1 fl(x)+ +Ap fp(x)I < e

forallxEX.

In other words, if a sequence of functions fn E Cp(X) does not exceed I in norm,

5. NAMIOKA'S THEOREM; AND PTAK'S APPROACH

11T

and converges in the topology of pointwise convergence to the function identically zero, then finite convex linear combinations of these functions uniformly approximate zero.

Proof. Consider the Cartesian product X x N+, and put U = {(x,n): Ifn(x) I > e}

and U(x) _ In E N+: (x, n) E U} for each x E X. Put also U(n) = {x. E X: (x, n) E U} for all n E N+. Denote by £ the family of all sets U(x), where x E X. We show that if g E M(N+, £, e), then I E{g(n) fn(x): n E N(g)}I < 2e for all x. E X. In fact, I E{g(n) fn(x): n E N+}I : E{g(n) ' Ifn(x)I: n E U(x)} + E{g(n) I fn(x)I: n E N+\U(x)} < E{g(n): n E U(x)}+eE{g(n): n E N+} < 2e. So, I E{g(n) fn(x): n E N(g)}I < 2e, i.e. E{g(n) fn(x): n E N(g)} is the linear combination looked for. It remains to prove that M(.N+, £, e) q' 0. Assume the contrary. Then by theorem 5.2 there are subsequences {nk: k E N+} in N+ (where nk, # nk" if k' 36 k") and {xk: k E N+} in X such that {n1, ... , nk} C U(xk) for all k E N+. Clearly, ni E U(xk) is equivalent to Xk E U(ni). Hence xk E (1(n1) fl fl U(nk) for every k E NI But. the set. (I(n) is closed, since fn is continuous. Thus, {U(n1): i E N I } is a centered family of closed sets in the compactum X. Consequently, there is an x' E fl{U(ni): i E N1J. But then fn;(x') > e for all i. E N1, contradicting the fact. that lim fn(x') = 0.

Remark. In the above given proof of theorem 5.3 we have only used the countable compactness of X and the upper semicontinuity of fn on X. Actually, theorem 5.3 remains true for arbitrary pseudocompact spaces, i.e. spaces on which every realvalued continuous function is bounded. A proof can be obtained by reduction to the case of compacta, which has already been considered.

We introduce some notation, to be used below.

Let Y be a compactum. Then C(Y) is the space of all realvalued continuous functions on Y, and p is the standard metric on C(Y): p(f, g) = max{I If (y) - g(y)I:

y E Y} for all f,g E C(Y). The closure of a set A C C(Y), in the metric space (C(Y),p), is denoted by cl,,(A), while 71 denotes the closure of A in C,,(Y) (and not

in RY!). Further, for e > 0 and g E C(Y), Of(g) = If E C(Y): p(f, g) < e} is the ball of radius e in (C(Y), p). The convex (algebraic) hull of a set A C C(Y) in the linear space C(Y) is denoted

by H(A); by definition, H(A) :_ {A1fl + n E N+}.

+ Anfn: Ai >- 0, E A, = 1, fi E A,

We need the following obvious corollary of theorem 5.3.

111.5.4. Proposition. Let Y be a compactum, A C C(Y), with g(Y) C [-1, 11 for all g E A, f E C(Y), and suppose that A contains a sequence {gn: n E N+} converging to f in the space CP(Y). Then f E clo(H(A)). Namioka's main result, from which the corollaries of interest to us and concerning compact sets of functions can be easily derived, is as follows.

ill. TOPOLOGICAL PROPERTIES OF FUNCTION SPACES

118

111.5.5. Theorem [125].

Let X and Y be compacta, and f: X x Y

Ra

realvalued function that is continuous in each variable separately.

Then there is an everywhere dense set A of type G6 in X such that f is jointly continuous in the two variables (i.e. with respect to the product topology on X x Y) at every point of the set A x Y.

Proof. We assume that l f (x, y) I < 1 for all (x, y) E X x Y. Define the map F: X - Cp(Y) by F(x)(y) = f (x, y). Since f is continuous in y, F(x) E Cl,(Y) for all x E X. The continuity of f in x implies that the map F: X --- CC(Y) is continuous.

The local oscillation of F at x E X is the nonnegative quantity

a(x) = inf{sup{p(F(xl),F(x2)) : X1, X2 E U} : U open in X and U 2) x}. If a(x) < 1/n, there is a neighborhood U of x in X such that, sup{p(F(xl), F(x2)): X1,x2 E U} < 1/n. Then also a(x') < 1/n for all x' E U. Thus, the set. Wn {x E X: cr(.r) < l/n.} is open in X, and the set A __ {x E X : ((x) = 0) n{11;,: n E N I } is of type G6 in X. Clearly, at all points of A x Y the function f is jointly continuous in its variables x, y. Therefore it suffices to prove that A is everywhere dense in X. Assume the contrary: let A 34 X. Since the space X has the Baire property, there .

is an e > 0 such that {x E X : a(x) < e} # X. For all x in the nonempty open set U = X \ {x E X : a(x) < e} we have a(x) > e. Then a(x) > e for all .r. E U (see above). Below we may assume, without loss of generality, that X = U (otherwise we replace the compactum X by the nonempty compactum U). So, it remains to derive a contradiction from the assumption that for some e > 0

we have a(x) > e for all x E X. We determine by induction a sequence l; = {U,,: n E N+} of open sets in X and a sequence 17 _ {xn: n E N+} of points in X such that the following three conditions hold for all n E N+: 1) xi E U;; 2) U;}1 C U1; and

3) for every x E U;+1 and g E Hi = H({F(x1),...,F(x;)}) (where ff is the convex hull of the set {F(xl),...,F(x1)} in C(Y)) the following inequality holds: p(F(x), g) ? 3

(*)

Put U1 = X, and take xl E X arbitrary. Suppose that the points xI,... , x E X have already been chosen, and that a nonempty open set U,, in X has already been determined. We show how to choose the point xn+l and set Un+1.

Clearly, the set H,, = HQ F(xl),...,F(xn)}) is a compact subset of the metric space (C(Y), p). Moreover, H,, is totally bounded in the metric p. This allows us to choose a finite set An C H,, such that p(g, A,,) < e/12 for all g E H,-

Put P(g) = {g' E C(Y): p(g,g') < 5e/12} for all g E A,,, and K. = U{P(g): g E An}.

If g" E C(Y) and p(g", H,,) < e/3, then p(g", An) < e/3 + e/12 = 5c/12, hence g" E K,,. Thus, if g" E C(Y) \ K. and g' E H,,, then p(g",g') > e/3.

5. NAMiOKA'S THEOREM, AND PTAK'S APPROACH

119

The definition of the topology of pointwise convergence trivially implies that the set P(g) is closed in Ci,(Y). Therefore its pre-image under the continuous map F is a closed set in X. We have F-'(P(g)) = {x E X: p(g, F(x)) < 5e/12} = T(g). For arbitrary xl,.r2 E T(g) we then have p(F(r,), F(x2)) < 5e/12 + 5e/12 < c. If X would contain a nonempty open set V lying entirely inside T(g), then for all x E V we would have a(x) < e, contradicting the fact that a(x) > e for all r. Thus, the closed set T(g) is nowhere dense in X for every gin the finite set An, and we can choose

a nonempty open set Un+, such that Un+, C Un and Un+I fl (U{T(g): g E An}) = 0. then F(x) U{P(g): g E An}, and hence p(g,F(x)) > e/3 for Now, if x E all g E Hn. We now take xn+, to be any point in Un+i The induction step of the construction is thus completed. We can thus assume that sequences 77 and e with the properties 1)-3) have been constructed. Properties 1)-2) and the compactness of X imply that some point. x of the set. d> = fl{Un: n E N"'} = fl{Un: n. E N+} is a limit point for the sequence 71 = {xn: Ti E N I ). Since F is cont.ilnms, F(cr) E l3, where 13 =_ { 1'(xn): n. E N"'-}.

Since X is compact and !% is continuous, we find that F'(X) = (F(x): x E X) is a compact subspace of C,,(Y). But Y is compact. Thus F(X) is a Fr&iici.--Urysohn space by theorem 3.6. By proposition 5.4 it now follows from F(j;) E F(X), B C F(X), and F(f) E B that F(f) E cl,,(H(B)). However, clearly II(B) = U{Hn: n e N+}, and i E d) C Un+I implies that p(F(x), Hn) > e/3. Thus p(F(r), H(B)) > e/3, contradicting F(f) E cl,,(II(B)). Theorem 5.5 has been proved. Using theorem 5.5 we can easily prove the following unexpected theorem on the relationship between the topologies of pointwise convergence and uniform convergence on compacta in sets of functions.

111.5.6. Theorem. Let Y be a compactum and Z a compact subspace of the space CP(Y). Then there is an everywhere dense set M of type G6 in Z such that at all points of M the topology generated on Z by the metric p (of uniform convergence, see above) coincides Tvith, the topology induced on Z by Cp(Y).

Proof. Every z E Z is a realvalued continuous function on Y. In correspondence with this, we define the map f : Z x Y -> R by f (z, y) = z(y) for all z E Z, y E Y. Then Z C CD(Y) trivially implies that f is separately continuous in each variable z, y. By theorem 5.5 there is an everywhere dense set M of type G6 in Z such that the function f is jointly continuous in its variables at all points of the set Af x Y. We show that M is the set looked for. Fix z` E M and e > 0. There is an open set C in Z x Y such that C D {z'} x Y and If (z, y) - f (z`, y) I < e for all (z, y) E G (this readily follows from the continuity

of f at the points of M x Y). Since Y is compact, there is an open set U 3 z' in Z such that U x Y C G. For all z E U we then have I z(y) - z (y) I = If (z, y) - f (z`, y) I < e

for all y E Y.

120

III. TOPOLOGICAL PROPERTIES OF FUNCTION SPACES

Since the topology generated by the metric p on Z is stronger, generally speaking, than the topology generated by pointwise convergence, we conclude that the two topologies coincide at all points of M.

111.5.7. Corollary [141]. Every Eberlein compactum contains an everywhere dense metrizable subspace.

Now we can easily obtain the following result of H. P. Rosenthal.

111.5.8. Theorem [141]. For every Eberlein compactum X the Suslin number and the weight coincide: c(X) = w(X).

Proof. Always c(X) < w(X) [66]. We put r = c(X) and prove that w(X) < T. The Eberlein compactum X contains an everywhere dense metrizable subspace Y by corollary 5.7. We have c(X) = c(Y) :5,r, and d(X) < d(Y). However, the Suslin number of a metrizable space equals its density. Consequently,

d(X) < d(Y) = c(Y) < r. The compactum X is monolithic by theorem 3.6. We conclude that nw(X) < d(X) < T. But w(X) = nw(X), since X is a compactum 166J.

So w(X) < r. Theorem 5.8 makes it possible to relate, in a rather unexpected manner, the Suslin number of a compactum X and the weights of compacts lying in C,,(X). More precisely,

111.5.9. Theorem (A. V. Arkhangel'skii [1]).

For any compact space X, the Suslin number of X equals the supremum of the weights of the compacta in Cp(X).

Proof. Put T = c(X) and A = sup{w(F): F C C,,(X) and F compact}. If ry is a disjoint family of nonempty open sets in X and p = 1ryl, then, as has been proved in §3 (proposition 3.2), CD(X) contains a compactum homeomorphic to the one-point compactification A,, of the discrete space of cardinality p. Since the weight of the compactum A,, is p, we conclude that r < A.

It remains to prove that A < -r. Let F C CC(X), F compact. The image of the compactum X under the canonical map tp: X - CD(F) is a compactum 4) which separates the points of F. Therefore, c(4)) < c(X) < T since i/i is continuous, and w(F) < w(4)) since F can be homeomorphically embedded in Cp(4)). However, 4) is an Eberlein compactum, hence, by theorem 3.8, w(4)) = c(4)) < r. We conclude that w(F) < w(4') < T, i.e. A < T.

111.5.10. Corollary [56].

If X is a compactum and X L Y, then c(Y) < c(X).

Proof. The inequality c(Y) < sup{w(F): F C Cp(Y) and F compact} is always fulfilled, while the equality c(X) = sup{w(F): F C Cp(X) and F compact} holds by theorem 5.9, since X is compact.

6. BATIJROV'S THEOREM

121

111.5.11. Corollary [56]. If X and Y are compacta and X L Y, then c(Y) = c(X). V. V. Tkachuk has generalized corollary 5.11 to the case when X and Y are (tech complete spaces. Theorems 5.5 and 5.6 can similarly be generalized. More precisely, preserving our

method of proof and the conclusion of the theorem, it is not difficult to generalize theorem 5.5 to the case when Y is a compactum and X is a Cech complete space. This also gives a direct generalization of theorem 5.6 (Y is assumed to be a compactum and Z a Cech complete space). Recall that if a (tech complete space has a countable network, then it has a countable base [66]. Having made the conclusions above, the proof of theorem 3.9 automatically becomes a proof of

111.5.12. Theorem [12]. Let. X be a compactum and Y a Cech complete suh. p ace of the space Cp(X). If the Suslin number of Y is countable, then Y is separable and metrizable.

Other applications of theorem 5.5 and its generalizations can be found in [125], Note that theorem 5.11 cannot be generalized to arbitrary spaces X and Y

[82].

(V. V. Uspenskii [63]).

6. Baturov's theorem on the Lindelof number of function spaces over compacta The main result of this section is in keeping with Grothendieck's theorem in §4, and even allows us to obtain a version of the latter. Nevertheless, it is completely unexpected.

Recall that the extent e(X) of a space X is the supremum of the cardinalities of discrete closed sets in X.

111.6.1. Theorem (D. P. Baturov). Let X be a Lindelof E-space. Then for every subspace Y of CD(X) the extent e(Y) of Y equals the Lindelof number 1(Y) of Y.

Proof. Always e(Y) < 1(Y). Hence it suffices to prove that if 1(Y) > T, then also e(Y) > -r. Without loss of generality we may assume that X is a Lindelof p-space, since every Lindelof E-space is a continuous image of such a space (see [123] and chapt. 0). Thus, we may assume that X allows a perfect mapping onto a space of countable weight. Suppose 1(Y) > r. There is an open cover y of Y in which there is no subcover of cardinality < r. We assume that the elements of y have the form

W(xa,...,xk;O1,...,Ok) = {f E CC(X): f(xi) E O., i = 1,...,k}, where xi are points in X and O; are elements of the standard countable base of the real number space R. It is further expedient to denote Wk(x; 0) = W (xi, ... , xk; 01, ... , Ok), where x = (XI, ... , xk) E Xk and 0 = Oi x . . . x Ok is an element of the

122

111. TOPOLOGICAL. PROPERTIES OF FUNCTION SPACES

base of F. The family -y can be represented in the form -y = U{-yn: n E N+}, where all Wk(x; 0) E yn correspond to one and the same number k = kn and one and the

same set 0 = On C W-. Put, for n E N+, An = {x E X k" : Wk" (x; O) E yn }. For every map f E Cc(X) and number k E N+ we denote by fk the map from Xk to Rk f (x)L.)). under which a point x = (x1,. .. , xk) becomes (f Using this notation, the fact that y covers Y can be written as: *) for each f E Y there are an n E N+ and x E An such that f A" (x.) E On. The fact that no subfamily of y of cardinality < r covers Y can be written as:

**) if B C An and IBn' < -r, where n E N+, then there is a g E Y such that

gk"(Bn)nOn=0 for allnEN+. We construct by transfinite recursion a set F = { fa: a < r+} C Y which is closed and discrete in Y. Choose fo E Y arbitrarily, and suppose that we are given an a < r+ and that for all fi < a functions f, E Y have been determined. Fix n E N+, and take a perfect map cbk" from Xk" onto some separable metrizable space Mn. f O .. . For each finite collection ni, ... ,.13,. < a we consider the map A f;,1A0k", the diagonal product of the maps f, , ... , f, and ¢k", defined on X'and with values in the space Rk-" x Mn, which is a space with a countable base. It is immediately seen that f(n# ..... #,) is a perfect map, for 0k" is perfect 1661. The space Rk"''' x Mn is hereditarily separable; therefore we can choose in An a countable subset S 1 . 6rl whose image ,q ......j3 )) is everywhere dense in Put B' = U{SSp....... ,l: 61...., fl < a}. Clearly, 1Bn1 <.-r. Hence (see **)) there is a function f,,, E Y such that f,,k,"(Ba) n On = 0 for all n E N+. The construction of the set F = f f,,,: a < r+} is finished. We show that F is discrete and closed in Y. Assume the contrary. Then there is a point g E Y which is a limit point for F, i.e. such that every neighborhood of g contains infinitely many functions from F. For some n E N+ and x E X'-" we have i.e. gk°(±) E On. The tightness of Cp(X) is countable (chapt. II, §1), gE and the tightness of Y does not exceed r. Hence there is an a' < r+ such that g is a limit point for the set {f,,,: a < a'). We let ao be the smallest such a' < r+, and put P = {fa E Wk" (x; On) : a < ao}.

The set (gk")-'(On) n n{(fk")-1(fk"(i)): f E P} contains x, and is thus nonempty.

Put T = n{(fk")-'(fk (x)): f E P) \ (g')-'(O,) We distinguish two situations.

Case 1. The set T is empty. Put 0"(1) = in. The map ¢k.: Xj'- -- Mn is perfect, hence 45k'(nz) is compact. Since the set (g")-'(On) is open, there is a finite set {fi,,..., f#,} C P such that

ID = n l(f;, )-'(f;, (x)) n Ok'(in) c (9k")-'(On) The set 4 is the complete pre-image of the point (f (±), ... , f (i), 9n) under the map f( ,...... ,) The map fps, ...... 6,l is perfect and, moreover, closed, while (gk")-'(On) is a neighborhood of the set '. Since 1 E A. n
6. RATUROV'S THEOREM

complete pre-image of some point of the set

123

..,a,)). Hence there is a point.

(O ). But then, by construction, the function that lies also in f ,, f o r a > max{pi: i = 1, ... , r}, does not belong to the neighborhood l'irk (.r.'; whereas the function g itself belongs to 61'k,,(x';0). Put a' = max{(I,: i = 1,...,r1. Clearly, a' < ao, and since g is a limit function for the set {f.: a < a0l, we conclude that g is also a limit function for the set (f,,: a < a'}, contradicting the choice of ao. Case 2. The set T is not empty. Take a point x" E T. Then gk (:r") # gk (s), since X, E

gk (x) E 0,,. At the same time f"- (x") = f'-(fl for all f E P. This implies that g does not belong to the closure of P, again a contradiction. So, the set F is discrete and closed in Y. Theorem 6.1 has been proved.

If X is a Lindelof E-space, then every countably compact Corollary. subspace Y of Cc(X) is a monolithic Prechet-Urysohn compactum.. 111.6.2.

Proof. Since. Y is conntahly compact., its extent is countable. I lence, by theorem (;. I, Y is a l,indeliif space. But then Y is compact [161. The tightness of CG(X) is countable and Cc(X) is monolithic, since X is a stable space (see theo(see theorem rems 11.6.8 and II.6.21). Hence Y is a monolithic compactum of countable tightness.

By proposition 3.7, Y is a F-echet-Urysohn compactum. The last result can be understood as a generalization in one direction of Crothendieck's theorem. In chapt. I, §5 we have established that if a space ,(X) is normal, then it has countable extent. Combining this theorem of Reznichenko with Baturov's theorem (6.1), we obtain the following result.

111.6.3. Theorem. If X is a Lindelof E-space, and

,(X) is normal, then CC(X)

is Lindelof.

111.6.4. Theorem.

Let X be a Lindelof E-space, g E CC(X), and CC(X) \ {g) a normal space. Then the space X is separable. Proof. Since the complement of Y = CC(X) \ {g} in C,(X) is finite and Y is normal, by theorems 1.5.12 and 6.1 Y is a Lindelof space. Consequently, {g) is a set of type G6 in C,,(X), i.e. C,,(X) is a space of countable pseudocharacter. By theorem 1.1.4, X is separable.

111.6.5. Corollary.

If X is a monolithic Lindelof E-space, and if the complement to some point in Cy(X) is normal, then X is a space with a countable network.

This follows from theorem 6.4 and the fact that every separable monolithic space has a countable network. In particular, if a compactum K is monolithic but not metrizable, then C,,(X) \ {g}

is not normal, for all g E CC(X). The following useful result, which is related to assertions 6.3, 6.4, and 6.5, has been obtained by D. P. Baturov: if X is a monolithic

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compactum of countable tightness, and if Y is an everywhere dense subspace of CP(X)

which is normal, then Y is a Lindelof space. It is worthwhile to compare this assertion with theorem 1.5.21, which was proved under the assumption of the continuum hypothesis.

111.6.6. Problem.

Let X be a compactum, and Y an everywhere dense subspace of C,(X). Is it then true that Y normal implies Y Lindelof?

CHAPTER IV

Lindelof number type properties for function spaces over compacta similar to Eberlein compacta, and properties of such compacta In this chapter we prove for a large class compacta that the function spaces over them have countable Lindelof number, or satisfy stronger requirements. A key role in many of the reasonings in this chapter is played by the Stone-Weierstrass theorem.

1. Separating families of functions, and functionally perfect spaces If X is a topological space, then E(X) denotes the class of all spaces that can be represented as a continuous image of the product of X and some compactum. A class P of topological spaces is called k-directed if the following conditions are

satisfied: 1) X, Y E P implies X x YEP; and 2) if X E P, then £(X) C P.

IV.1.1. Examples.

The class of all compacta is k-directed. Clearly, the class of all o-compact spaces and the class of all Lindelof E-space are also k-directed.

IV.1.2. Proposition.

Let P be a k-directed class of spaces, Y a space, y E Y, and

Y= Y \ {y} EP. Then also Y EP. In fact, Y is a continuous image of the space Y x {O,1}, which clearly belongs to P.

The system of notions developed above `works' in relation to the following construction. Below, X is some space, n E N+, In = [-n, n] is a closed interval in R. If

A, B C Cp(X ), then 01(A, B) = {max{ f, g}: f E A, g E B}, 02(A, B) = {minIf , g}: f E A, g E B}, and On(A) = {a f + bg: a, b E I,,, f, g r= A}. The subspaces ?P1 (A, B) and 02(A, B) of CC(X) are continuous images of the space A x B; On(A) is a continuous

image of the space A x In x In.

Let Y C Cp(X). Put S1(Y) = {Y}, and put for k E N+, Sk+1(Y) = Sk(Y) U {,il(A, B): A, B E Sk(Y), i E {1, 2}} U {0,(A): A E Sk(Y), n E N+}. In this inductive way a family SS(Y) is defined, n E N+. It can be readily verified by induction that every Sn(Y) is countable. Moreover, if P is a k-directed class of spaces and Y E P, then it can be readily proved by induction that S,,(Y) C P for all 125

IV. LINDELOF NUMBER TYPE PROPERTIES FOR FUNCTION SPACES

126

n E N. This gives the following conclusions concerning the family S(Y) = n E N+}.

IV.1.3. Proposition. If P is a k-directed class of spaces, and Y C Cc(X), Y E P, then the family S(Y) is countable, and S(Y) C P. The notation S(Y) is preserved in the sequel. We now state and prove the basic technical result.

IV.1.4. Proposition. Let X be a compactum, Y C C,,(X), e E Y (where e(x) = 1 for all x E X), and suppose Y separates the points of X. Then the set M = US(Y) is everywhere dense in the space C(X) endowed with the topology of uniform convergence

on X. Proof. The definition of S(Y) implies that if f, g E Af and a, b E R, then max f f, g} E Af, min{f, g} E M, and of +bg E M. Moreover, r. E Y C Al, and since Y separates

the points of X, M also separates the points of X. It remains to apply the StoneWeierstrass theorem in Kakutani's form 166, theorem 3.2.21). In its full generality, proposition 4.1 will only be used in the next section. Below we derive only those simplest consequences of proposition 1.4 in which it suffices to know that Al is everywhere dense in C,,(X). We also need the following generalization of a weaker version of proposition 1.4.

IV.1.5. Proposition.

Let X be an arbitrary space, Y C C,,(X), e E Y (where e(x) = 1 for all x E X), and suppose Y separates the points of X. Then the set M = US(Y) is everywhere dense in the space Cc(X), endowed with the compact-open topology, and is thus everywhere dense in CC(X).

A proof of this assertion can be obtained if in the proof of proposition 1.4 X is not taken to be compact, and the reasoning is restricted to arbitrary compact subspaces of X.

IV.1.6. Proposition.

Let X be a compactum. Then the following conditions are

equivalent:

a) X is an Eberlein compactum; b) there is a compactum F C CC(X) separating the points of X; c) there is a o-compact subspace Y C CC(X) separating the points of X. Proof. Clearly, b)=:>c). We prove c)=*-a) and

c)=a). The evaluation map ': X -- Cc(Y) is continuous, and distinct points remain distinct, since Y separates the points of X. Since X is a compactum, ip homeomorphically maps X onto the compactum tp(X) C C,,(Y). But bb(X) is an Eberlein compactum, since Y is or-compact (see theorem III.1.11). Hence X is an Eberlein compactum.

I. SCPARATING FAMILIES OF FUNCTIONS

127

Let X be an Eberlein compactum, i.e. X C Cp(Y), with Y a compactum. Let F be the image of Y under the evaluation map i/J: Y -+ Cc(X). Clearly, F is compact and separates the points of X. A space Y is called k-separable [12] if Y contains an everywhere dense a-compact subspace.

IV.1.7. Theorem.

A compactum X is an Eberlein compactum if and only if the space Cp(X) is k-separable.

Proof. If Y is an everywhere dense o-compact subspace of C'(X), then Y separates the points of X, and by proposition 1.6 X is an Eberlein compactum. This proves the sufficiency.

Necessity follows from propositions 1.6 and 1.4: if a compactum Y C Cc(X) separates the points of X, then the space M = US(Y) is o-compact (see proposition 1.3) and everywhere dense in C,(X).

IV.1.8. Corollary.

If X is an Eberlein compactum, and a compactum Y is 1equivalent to X, then Y is also an Eberlein compactum.

In relation with proposition 1.6 it is appropriate to consider the following notions. A space X is called functionally perfect' if CC(X) contains a compactum F separating the points of X. Clearly, the functionally perfect compacta are precisely the Eberlein compacta.

In turn, theorem 1.7 provides a reason for the isolation of the class of dually-kseparable spaces: spaces X for which Cc(X) is k-separable. Proposition 1.5 and the fact that if Y is a compactum, then A1 = US(Y) C C'(X) is a o-compact subspace, imply

IV.1.9. Proposition.

If it space X is functionally perfect, then Cc(X) is k-

separable.

On the other hand we have

IV.1.10. Proposition.

If G ,(X) contains a o-compact subspace separating the points of X, then X is functionally perfect. The proof of this is simple: it is the same as that of assertion 111.1.11. Propositions 1.9 and 1.10 imply

IV.1.11. Theorem. A space X is functionally perfect if and only if it is dually-kseparable, i.e. if and only if Cp(X) is k-separable.

This fact was noted by M. Tseitlin and V. V. Tkachuk. Theorem 1.11 generalizes 1 Such spaces are also called functionally . omplete spaces (see /1/); in /sl they are called functionally perfect.

IV. LINDELOF NUMBER TYPE PROPERTIES FOR FUNCTION SPACES

128

theorem 1.7.

IV.1.12. Corollary. If X

e

Y and X is functionally perfect, then Y is functionally

perfect.

IV.1.13. Proposition.

If a space X is k-separable, then CP(X) is functionally

perfect.

Proof. Under the canonical evaluation map i/i: X --+ CPCP(X) the space X is homeomorphically mapped onto the subspace p(X) C CPCP(X), which separates the points of CP(X). The space -O(X) contains an everywhere dense o-compact subspace, also separating the points of CP(X). By proposition 1.10 we conclude that CP(X) is functionally perfect. In distinction to proposition 1.9, proposition 1.13 has no converse.

IV.1.14. Example (V. V. Tkachuk).

Let X = {a: a < w1} be the space of

all ordinals smaller than the first uncountable ordinal w1, with the ordinary topology.

Clearly, X is not k-separable (since X is not separable and all compacta in X are countable). We show that CP(X) is functionally perfect. We first note the following two facts.

1. Let f, g E CP(X), g(0) = f (0), and f (a + 1) - f (a) = g(a + 1) - g(a) for all a < w1. Then f = g. In fact, if f 0 g then the smallest ordinal a' < w1 for which f (a*) g(a') cannot he a limit ordinal, by the continuity of f and g. But a* can also not be an ordinal of the form a + 1, since in this case we would have f (a) 9-` g(a), contradicting the choice of a`.

II. For each a < w1 and each f E CP(X) we put ga (f) = f (a + 1) - f (a). Then g0, E CPCP(X ), and for every f c: CP(X) and e > 0 the set of all a < w1 for which Jga(f) > e is finite. The first assertion is obvious. The second readily follows from the continuity of f and the definition of the topology on X. Take the function h E CPCP(X) defined by h(f) = f (O) for all f E CP(X ), and denote by 0 the function on CP(X) that identically vanishes. Put F = {h} U {gQ: a < w1 } U {0}. By II, F C CCP(X ). The second part of II implies that each neighborhood of 0 in CCP(X) contains all elements of F except finitely many. Thus F is a compact space (the one-point compactification of a discrete space). Assertion I implies that F separates the points of CP(X). Hence CP(X) is functionally perfect.

It remains unclear whether k-separability of a space X can be characterized by a topological property of CP(X).

IV.1.15. Problem. Let X -t Y, and let X be k-separable. Is it then true that Y is k-separable?

1. SEPARATING FAMILIES OF FUNCTIONS

129

We note the simple

IV.1.16. Proposition. If a space X is functionally perfect, then every subspace Y of X is also functionally perfect.

Proof. Under the map of restricting a function f E C,,(X) onto a subspace Y, a compactum F C C,,(X) separating the points of X is mapped onto a compactunn in CP(Y) separating the points of Y. Despite the lack of a 'dual' duality between being functionally perfect and kseparable, the following facts hold.

IV.1.17. Proposition [12]. The space CPCC(X) is functionally perfect if and only if the space CC(X) is k-separable.

Proof. Let CCp(X) be functionally perfect. Since X is homeomorphic to a subspace of CCI,(X), proposition 1.16 implies that X is functionally perfect. Hence Cp(X) is k-separable, by proposition 1.9. Conversely, if C,(X) is k-separable, then CCc(X) is functionally perfect by proposition 1.13.

IV.1.18. Theorem. If a space X is functionally perfect, then C,,2.-I(X) is k-separable for all n E N+, and Cp,2,,(X) is functionally perfect for all n E N+. Conversely, if C,2,_1 (X) is k-separable or Cp,2,, (X) is functionally perfect for some n E N, then X is functionally perfect.

Proof. This follows immediately from theorem 1.11 and propositions 1.13 and 1.17.

IV.1.19. Corollary. If a space X is functionally perfect and X n E N, then Y is also functionally perfect.

Y for some

IV.1.20. Example. Let a topological space X have the form X = U{I,,,: a E A}, where each I,,, is homeomorphic to an interval, n{Ia,: a c= A} = {0}, while the family {II \ {0}: a e A} is an open disjoint cover of the space X \ {0} (such a space is called 'hedgehog-like'). For each a E A we denote by f,, some realvalued function on X which homeomor-

phically maps the interval Ia onto some interval in R., where 0 is mapped to 0 and all intervals I& with a' a are mapped to 0. The assumptions concerning X make it clear that f,,, is continuous. Put F = { fa: a E Al U {6}, where 6 is the function on X that vanishes identically. Clearly, each neighborhood of 0 in Cc(X) contains all elements of F except finitely many. Hence F is a compactum (of the form Ar, where r = JAS). Clearly, the family f f,,,: a E A), and with it the compactum F, separates the points of X. Hence X is functionally perfect. This example is extremely important, as we will see below.

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IV. LINDELOF NUMBER TYPE PROPERTIES FOR FUNCTION SPACES

IV.1.21. Proposition [1].

A product X = jj{ X,,: n E N} of countably many

functionally perfect spaces is functionally perfect.

separating the points of X,,. The image Proof. Let F,, be a compactum in of F under the map urn: Cc(X,,) - C9(X) dual to the projection map ir,,: X X is denoted by F,,. Then Dn is compact and, as can be readily seen, the subspace Y = U{4,,: n E N) of C'(X) separates the points of X. Proposition 1.10 now implies that X is functionally perfect.

IV.1.22. Theorem. Every metrizable space is functionally perfect. Proof. The metric Kowalsky hedgehog of arbitrary cardinality is functionally perfect, since it belongs to the class of 'hedgehog-like' spaces (see example 1.20, and [661). By proposition 1.21, the countable power of the Kowalsky hedgehog is a functionally perfect space. However, by Kowalsky's theorem, each metrizable space is homeomorphic to a subspace of the countable power of the Kowalsky hedgehog. Proposition 1.16 now implies that each metrizable space is functionally perfect. Another widening of our views concerning the class of functionally perfect, spaces is made possible by

IV.1.23. Proposition.

If a space X can be condensed onto a functionally perfect space Y, then X is itself functionally perfect.

Proof. Under the map dual to the condensation f : X -+ Y, the space C1,(Y) is homeomorphically mapped onto an everywhere dense subspace of CP(X) (see chapt. 0). subspace, Cp(X) contains such Since Cp(Y) contains an everywhere dense a subspace too.

IV.1.24. Corollary.

Every paracompactum with G6 diagonal is a functionally

perfect space.

Proof. This follows from assertions 1.22, 1.23 and the fact that any paracompactum with G6 diagonal can be condensed onto a metrizable space [16[, [66]. Theorem 1.22 can be derived also from the following useful result.

IV.1.25. Theorem (A. V. Arkhangel'skii).

Each metrizable space is homeo-

morphic to a subspace of some Eberlein compactum.

Proof. By theorem 3.4 and Kowalsky's theorem, it suffices to prove that a metric Kowalsky hedgehog of arbitrary cardinality can be topologically embedded in an Eberlein compactum. This we do as follows. The discrete space DT of cardinality r can be embedded in the Eberlein compactum A, (see 111.3.3). The product D, x I can be embedded in the product A, x I, where

2. SEPARATING FAMILIES OF FUNCTIONS ON COMPACTA

131

I = 10, 11. Fix a compactum F (= C,(A,) separating the points of A,. We may assume

(see chapt. 0) that if (x)l < 1 for all f E F and all x E A,. For each f E F we define the function f E CP(A, x I) by f (x, A) = (1 - A) f (x) for all (X,,\) E A, x I. We also piit h(x, A) = 1 - A for all (x, A) E A, x I. Clearly, h and f are continuous, and the family F' _ If: f E F} U {h} separates all points of A, x I except the points of the form (x', 1), (x", 1). It is also clear that the subspace

F'CCP(A,x1)isacompactum,andthat g(x,1)=0forallxEA,andallgEF'. We now glue together the 'upper base' of the product A, x I, i.e. the set {(x,1): x E A,}, into a point. The space A, x I is then mapped onto a compactum Y, and F' is contained in the image of CP(Y) under the map dual to this glueing map 40. This means that CP(Y) contains a compactum homeomorphic to F' and separating the points of Y. Hence Y is an Eberlein compactum. The subspace c(V, x I) of Y is a topological copy of the metric hedgehog of cardinality T. This follows since A, and I are compact. So, we have proved that a inetrizable hedgehog of arbitrary cardinality is homeomoihhic to a subspace of an Eberlein compactum. Theorem 1.25 has been proved. C. Dimov has given an 'intrinsic' characterization of those spaces that. can be en)bedded in Eberlein compacta.

2. Separating families of functions on compacta and the Lindelof number of C,(X) In this section we need the following definitions and notations.

V, denotes the discrete space of cardinality r, with r an infinite cardinal. The space (X x D,)7 = XT x (D,)T is called the r-hull of X, and is denoted by o,(X). If X is homeomorphic to o,(X ), we say that X is r-invariant. The following assertions are obvious.

IV.2.1. Proposition.

The space o,(X) is r-invariant for any space X (and any

infinite cardinal r).

IV.2.2. Proposition. If a space X is r-invariant, then the spaces XT and X x D, are hom.eomorphic to it.

Applying proposition 2.2 we obtain

IV.2.3. Proposition. Let a space X be r-invariant. Then: a) if IMI < r and Xa E E(X) for all a E M, then the Tikhonov product jj{Xa: a E M} also belongs to £(X); b) if -y is a family of subspaces of a space Y and 1-yj < r, -y c E(X), then. U-y E E(X).

In this section, a key role is played by proposition 1.4 from §1.

If P is a class of spaces, then (P)6d denotes the family of spaces X that can be represented as X = fl(X,: i E N+}, where each Xi is the union of a countable family

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IV. LINDELOF NUMBER TYPE PROPERTIES FOR FUNCTION SPACES

of spaces from P. The following assertion is the main general theorem of this section.

IV.2.4. Theorem.

Let X be a compactum, and P a k-directed class of spaces.

If there is a Y C CC(X) such that Y E P and Y separates the points of X, then Cp(X) E (P)06 Proof. By proposition 1.2, the subspace Y = Y U {e} of Cp(X) (where e is the function identically equal to 1 on X) belongs to P. Put M = US(Y), and for n E N,

n'In = {g E Rx: there is an f E M for which Ig(x) - f(x)I < 1/n for all x E X}. By proposition 1.4, Cp(X) C M' = fl{M,,: n E N}. On the other hand, M' C Cp(X), since the limit of a uniformly converging sequence of continuous functions is a continuous function. Thus fl{Mn: n E N} = CC(X). But the subspace Mn of RX (endowed with the topology of the Tikhonov product) is clearly the continuous image under a map h of the space M x (111.)X, where (11/n)' is the Tikhonov product of

IXI copies of of the ordinary interval I1/n = [-1/n, l/n]. Since Y E P and P is kdirected, S(Y) C P by proposition 1.3. Thus, F x (II/n)X E P and h(F x E

P for all F E S(Y). We find that Mn = h(F x (111.)X) = U{h(F x (111n)x): F E S(Y)} is the union of a countable family of spaces in P (since S(Y) is countable, see proposition 1.3). So, CC(X) = fl{Mn: n E N} E (P)a6. Recall that a space Z is called a space of type Kb if in some ambient space it is the intersection of a countable family of o-compact spaces, i.e. spaces each of which is the union of countably many compacta. The following theorem was first established by M. Talagrand [147] in a somewhat weaker formulation (see also [148], [69], [25], [2]).

IV.2.5. Theorem. If X is an Eberlein compactum, then Cp(X) is a space of type K36

Proof. There is a compactum F C CC(X) separating the points of X, by proposition 1.6. Since the class 1C of compacta is k-directed, theorem 2.4 implies that CC(X) belongs to the class ()C)o6. Similarly, every space of type Kb is also a Lindelof E-space [3]. Hence we have

IV.2.6. Corollary. For every Eberlein compactum X, the space CC(X) is a Lindelof E-space.

M. Talagrand [149] has shown that theorem 2.5 has no converse; we will discuss this result somewhat later. On the other hand, R. Pol [134] has established that if X is a scattered compactum and Cp(X) is IC-analytic (in particular, if it is of type Ka6), then X is an Eberlein compactum. We now dwell on some other general results, allowing us to obtain theorem 2.4.

IV.2.7. Theorem. Let X be a compactum, Y C Cc(X), and suppose Y separates

2. SEPARATING FAMILIES OF FUNCTIONS ON COMPACTA

133

the points of X. Then there are a, compactum P and a closed subspace B in oN0(Y) x P

such that CP(X) is a continuous image of B.

Proof. The class P = E(oN0(Y)) is k-directed, since oN0(Y) is Bo-invariant (see propositions 2.1 and 2.3). By theorem 2.4, Y E P implies that CP(X) E (P),6. But if -y C P and'-y countable, then Ury E P by proposition 2.3. Thus there are Z, E P such that CP(X) = fl{Z1: i E N+}. But then Cp(X) is homeomorphic to a closed subspace of the topological product T = 11{Z=: i E N+}. By proposition 2.3, T E P. It remains to recall the definition of P. A class P of spaces is called r-perfect (where r is an infinite cardinal) if the following conditions hold for any X E P: 1) O1(X) E P;

2) E(X) C P; 3) if Y C X and Y is closed in X, then Y E P.

IV.2.8. Proposition. The class of all Lindelof E-spaces is lto-perfect. This readily follows from the definition of Lindelof E-space. At the same time, theorem 2.7 gives the following conclusion.

IV.2.9. Theorem.

Let X be a compactum, and P an Bo-perfect class of spaces. Then CP(X) E P if and only if there is a Y C CP(X) such that Y E P and Y separates the points of X. This and proposition 2.8 imply

IV.2.10. Corollary. Let X be a compactum. Then CP(X) is a Lindelof E-space if and only if there is a Y C CP(X) which separates the points of X and is a Lindelof E-space.

IV.2.11. Corollary.

Let X be a compactum. Then the following conditions are

equivalent:

1) CP(X) is a Lindelof E-space; 2) some everywhere dense subspace of CP(X) is a Lindelof E-space. Of course, theorem 2.9 implies a more general result.

IV.2.12. Corollary. For any compactum X and any lZo-perfect class P of spaces the following conditions are equivalent: 1) CP(X) E P;

2) there is a Y C CP(X) such that Y = CP(X) and Y E P. The following a: rtio_^. i5 d al to theorem 2.9.

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IV. LINDELOF NUMBER TYPE PROPERTIES FOR FUNCTION SPACES

IV.2.13. Theorem. Let P be an l1o-perfect class of spaces, X E P, and Y C CP(X ), Y compact. Then Cp(Y) E P.

Proof. The image ?P(X) of X under the evaluation map I): X -+ CP(Y) separates the points of Y, and obviously VI(X) E P. Thus, by theorem 2.9, CP(Y) E P. These results can be given a slightly more general statement, but with less obvious range of applications. A class P of spaces is called k,6-directed if it is k-directed and closed under the operations of taking the union of a countable family of spaces and the intersection of a countable family of subspaces. Recall that k-directedness of a class P is equivalent to P containing all compacta, the product of two arbitrary spaces in P, and the image of a space in P under a continuous map. The reasoning given above makes it clear that every l' o-perfect class is k,,6-directed. Theorem 2.4 implies an assertion which is formally stronger than theorem 2.9.

IV.2.14. Corollary. Let X be a compactum, P a k,6-directcd class of spaces, and suppose there exists a subspace Y C Cp(X) separating the points of X and such that Y E P. Then also CP(C) E P.

IV.2.15. Example.

Let r be an infinite cardinal, D, the discrete space of cardinality r, C V DT, and LT = D. U a space in which only the point i; is not isolated, and where the neighborhoods of l; are all sets V C LT such that f E V and LT \ V is countable. 'Clearly, LT is a Lindelof P-space.

Let f : L, -+ R be an arbitrary function. It is obvious from the definition of topology in L, that the conditions f E CP(L,) and f (f) = 0, taken together, are equivalent to the countability of the set {x E L,: f (x) 0}. This makes it clear that 0} of Cp(L,) is homeomorphic to the subspace the subspace {f E CP(L,): f E RT of RD' = RT consisting of all points for which the set of nonzero coordinates is countable (possibly, finite or empty). Compacta in ERT (or homeomorphic images of them) are called Corson compacts [119]. Thus, every Corson compactum can be topologically embedded in CP(L,), for a certain r. For each cardinal T > Rio, let M, be the smallest class of spaces satisfying:

1) L, E M,; 2) every compactum belongs to M,; 3) if X E M, and Y C X, Y closed in X, then Y E M,; 4) the product of any countable family, of spaces in M, belongs to M,; 5) the image of a space in M, under a continuous map belongs to M,. We clearly have

IV.2.16. Proposition. For any T > No, M, is an l o-perfect class of spaces. Theorem 2.13 and proposition 2.16 imply

IV.2.17. Proposition. If Y C Cp(L,) and Y is compact, then CP(Y) E M,.

2. SEPARATING FAMILIES OF FUNCTIONS ON COMPACTA

I3S

Our nearest purpose is to prove that all spaces in M, are Lindelof. The following assertion can be proved without difficulty.

IV.2.18. Proposition. The class .Mr consists of all spaces that can be represented as a continuous image of a closed subspace of the product of the space (L,)' and a compactum.

Since the Lindelof number of a space does not increase under multiplication of the space by a compactum, it now suffices to establish

IV.2.19. Proposition. For any r > No the space (L,)"' is a Lindelof space. The proof of proposition 2.19 is divided into several steps. A space Y is called completely screenable if we can inscribe in any family y of open sets in Y a or-disjoint family it of open sets such that U-y = Ult. We first show that X = (L,)`° is a completely screenable space.

Put X. = (L,)", and let pn denote the natural projection from X onto X. Then Xn+I = X. x (L, \ { f }) U (X, x { f } ), where L, \ {t; } is a discrete space and Xn x {l; } is a retract of Xn+I. This implies that if Xn is completely screenable, then so is X"+I.

But XI = L, is clearly a completely screenable space. Consequently, all spaces Xn are completely screenable. We now consider an arbitrary family ry of open sets in X = (L,)`'. We may assume

that all elements of -y belong to the standard base of the product (L,)'. Then -y =

U{-y.: n E N+}, where 'yn c {p,-,'(U): U C Xn, U open in Xn}. Since X" is completely screenable, there is a a-disjoint family it,, of open sets in Xn such that {pn'(U): U E p,) is inscribed in -y" and U'yn = U{pn'(U): U E ltn}. Then 17 = {pn'(U): U E ltn, n E N+} is it a-disjoint family of open sets in (L,)'', inscribed in 'y, and U-y = Url. Hence (L,)' is completely screenable. This clearly implies that the space (L,)" (as any open subspace of it) is metaLindelof, i.e. in every open cover of this space we can inscribe a point-countable open cover. On the other hand, there is the well-known

IV.2.20. Lemma. For any open cover ry of an arbitrary space X there is a discrete

set ACX suchthatX=U{UEy: UflA340}. Hence, to prove that (L,)' is a Lindelof space, it suffices to prove that every discrete sets A in (L,)"' is countable. Clearly, without loss of generality we may assume that T=1kl. Below we prove a somewhat more general assertion, which we will need in the sequel.

We say that a space E has the strong condensation property if for each uncountable

subset A C E there is an uncountable subset C C A which is concentrated near a point c of E, i.e. such that the set C \ V is countable for every neighborhood V of c. We have

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IV. LINDELOF NUMBER TYPE PROPERTIES FOR FUNCTION SPACES

IV.2.21. Proposition. Let E be a space of weight < 111 having the strong condensation property. Then its countable power E" is a Lindelof space.

Proof. Since the weight of E" does not exceed ltli it suffices to show that every set A C E" of cardinality RI in E" has a complete accumulation point. So, let A C E''' and JAS = X I. We denote by pn the map from E" onto E under which an arbitrary point x E E" is sent to its nth coordinate x,,. We will successively determine uncountable sets Ao = A D AI J . and points c1i C2.... E E such that pn either sends all of A to the single point cn, or sends distinct points in A. to distinct points and such that pn(An) is concentrated near cn. Suppose that A. has been constructed, and consider the set B = pn+I(A,,). If B is uncountable, there are an uncountable set C C B and a point cn+i E E such that C is concentrated near cn+1. In this case we define A,,+1 by taking one point in each set pn+I (c) n An, where c runs through C. If B is countable, we can choose a point cn+I such that the set p;+1(cn+1)nAn is uncountable. We then put An+1 = pn+1(cn+1)nAn We show that the point c = {cn: n E N+} E E" is a complete accumulation point for A. Let U = U1 x x Uk x E x . . . be an arbitrary basic neighborhood of c. By construction, for each n < k the set T,, of all points of Ak whose images under pn do not belong to U,, is countable (possibly, empty). Hence the uncountable set Ak \ U{Tn: n < k} is contained in U. So, c is a complete accumulation point for A. Proposition 2.19 has been proved. As already noted, proposition 2.19 implies that all spaces in the class MT are Lindelof. Proposition 2.17 and example 2.14 now imply

IV.2.22. Theorem. For every Corson compactum X the space CP(X) is Lindelof. This result was obtained by S. P. Gul'ko [24] and K. Alster and R. Pol [69). The converse of 2.22 is not true: there is a compactum X which is not a Corson compactum but for which CC(X) is a Lindelof space [133].

3. Characterization of Corson compacta by properties of the space C,,(X) One of the fundamental results in this section is as follows.

IV.3.1. Theorem. A compactum X of weight <,r is a Corson compactum if and only if the space Cp(X) is a continuous image of a closed subspace of the space (L,)"

This theorem will be derived from certain auxiliary assertions. Below, D = {0,1 } is the discrete colon, regarded as an ordered group with respect to addition.

IV.3.2. Lemma.

Let X be a compactum, M C CC(X,D), where M separates the points of X and satisfies the condition: if f,g E M, then max{ f, g} E M, min{ f,g} E

M, and f +1 E M (where (f + 1)(x) = 0 if f (x) = 1; (f +1)(x) = 1 if f (x) = 0). Then M = C(X, D).

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Proof. An arbitrary continuous function f E C(X, D) is characterized by the openclosed subset Uf = f-1(0) of X. The family BM = {Uf: f E M} has the properties: 1) if V E BM, then X \ V E BM; 2) if V1, V2 E BM, then Vl fl V2 E BM; 3) if 141, V2 E BM, then V1 U V2 E BM;

4) for two arbitrary distinct points x, y E X there is a V E BM such that x E V,

y¢ V. This follows immediately from the assumptions in lemma 3.2. Since X is a compactum, properties 1)-4) of the system BM of open sets in X imply that every open-closed set in X' belongs to BM. But this means that C(X,D) = M.

IV.3.3. Proposition. Let P be a class of spaces satisfying: a) if Y E P, then every continuous image of Y belongs also to P; and b) if Y, Z E P, then also Y X Z E P.

If X is a compactum and there exists a Y C Cc(X, D) such that Y E P and Y separates the points of X, then CI,(X,D) E Pa, i.e. CI,(X,V) can be covered by a countable family of subspaces of it which belong to the class P.

Proof. We define by induction a family S of subspaces of Cp(X, D), for every n E N.

Let S1 = {Y}. If the family Sn has already been defined, then we put Sn u {o1(A x B): A, B E Sn, i = 1, 2} U {q5(A): A E Sn}, where

V), (A x B) _ {min{ f, g} : f E A, g E B);

V'2(AxB)_{max{f,g}: f EA,gEB};

ci(A)={f+1: f EA}. The maps ?PI, ib2, and ¢ are continuous, hence the assumptions concerning P and the

fact that Y E P imply by induction that Sn C P for all n E N. It is also proved by induction that all families Sn are countable. But then so is the family S', while the subspace M = US' belongs to Po (here S' = U{Sn: n E N}). The definition of M makes it clear that M satisfies the assumptions of lemma 3.2. Hence M = C(X, D), and C(X, D) E Pa. The class of all continuous images of all possible closed subspaces of the space (L,)'

is denoted by M°. If a space Y belongs to the class M° for a certain r, then Y is called a primary Lindelof space (of level r). In correspondence with proposition 2.19, all spaces in a class M° are Lindelof spaces. The space L, contains an infinite closed discrete subspace. This and the definition of M° readily imply

IV.3.4. Proposition. The union of a countable family of primary Lindelof spaces is a primary Lindelof space. The class of primary Lindelof spaces is closed under the operation of taking a (countable) product, transition to a continuous image, and transition to a closed subspace.

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If X is a zero-dimensional compactum, and CC(X,D) IV.3.5. Proposition. contains a primary Lindelof space Y separating the points of X, then Cp(X) is a primary Lindelof space.

The proof of proposition 3.5 is based on two lemmas.

Let X be a zero-dimensional compactum. Then there is a IV.3.6. Lemma. continuous map ¢ from the Cantor perfect set D"0 into the interval I such that for each continuous map f : X -+ I there is a continuous map g f : X --+ D"0 for which

f =q5ogf Proof. For each f E C(X, I) we form the space X f = X x If}, denote by e f the canonical homeomorphism from X to X f given by ef(x) = (x, f) for x E X, and consider the free sum Z = E0{X f: f E C(X, I)} of the spaces Xf. The space Z is paracompact and zero-dimensional: dim Z = 0. Therefore its Stone-Cech compactiflcation 07 is a zero-dimensional comps eLnm.

Consider the map h: Z --+ I for which h(x,f) = f (X) for all x E X and all f E C(X, I). The map h is continuous, hence its continuous extension h onto /3Z is defined, h: /3Z -+ I. According to the well-know factorization lemma of S. Mardesic (66], there are a zerodimensional compactum K of countable weight and continuous maps s: 6Z K and u: K --+ I such that h = uos. We assume that K is a (closed) subspace of the Cantor perfect set D"O. Then it can be extended to a continuous map v.: D"0 --+ I, and the equality h = u o s holds. We show that ¢ = u: D"° -+ I is the map looked for. Take an arbitrary f E C(X,1), consider the canonical map of from X to X f = X x If } C Z, where ef(x.) = (x, f )

forallxEX,andputgf=soef: X-+KCD"0. Then 4ogf=uosoef= f, i.e. the map gf: X -+ D140 satisfies the requirements of lemma 3.6.

IV.3.7. Lemma.

If X is a zero-dimensional compactum, then the space Cp(X, I) is a continuous image of the space Cp(X, Dk0).

Proof. Fix a map ¢: D"° -+ I as in lemma 3.6. Define the map p: Cp(X,D"0) --> Cp(X, I) by p(g) = .0 o g for all g E Cp(X, D'0). Lemma 3.6 implies that p maps Cp(X, D"0) onto CC(X, I). It is obvious that p is continuous.

Proof of proposition 3.5. Propositions 3.3 and 3.4 imply that the space CC(X,D) is primary Lindelof. But then so is the space (Cp(X, D))"°. Hence the space Cc(X, D"°) is primary Lindelof, being homeomorphic to (CC(X,D))"0 (see chapt. 0). By lemma 3.7 (and proposition 3.4), the space Cp(X, I) is primary Lindelof. But since X is a com-

pactum, every continuous function on it is bounded on it, and hence Cp(X) is the union of a countable family of spaces homeomorphic to Cp(X, I). Applying proposition 3.4 again, we conclude that Cp(X) is a primary Lindelof space.

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IV.3.8. Corollary. If a compactum X lies in Cp(LT, D), then Cp(X) is a primary Lindelof space.

Cp(X,D) the image i/'(LT) is a primary Proof. Under the evaluation map t/,: LT Lindelof space separating the points of X. The space Cp(LT,D) is zero-dimensional, hence X is a zero-dimensional compactum. Proposition 3.5 now implies that Cp(X ) is a primary Lindelof space. One step remains in the proof of the necessity part of theorem 3.1.

IV.3.9. Proposition.

Every Corson compactum of weight < r is a continuous

image of some compactum in CC(LT,D).

Proof. The space CP(LT,D) clearly contains a subspace Y which is homeomorphic to the E-product of r copies of the discrete colon D = {0,1}. But Y is also homeomorphic

to the E-product of r copies of the Cantor perfect set C. We assume that. 0 E C. 'T'here is a catltitt1t<m5 1111th 4/i: C - I such that 0((,') == I an(l (0} = 45_I(0).

Consider now an arbitrary Corson compactum B of weight < r. The conthulann B may be assumed to lie in Z = Ell(1a: a E A}, the E-product of the intervals 1a,

where JAI =rand I,=Ifor all aEA. Put Ca=Cand4, =0:C,--lIforall

a e A, and consider the E-product Zo = E fj{Ca: a E A} of CAI = r copies of the Cantor perfect set Ca = C. Define the map g from Zo = fl{Ca: a E A} to Z = jj{Ia: a E Al by: if z = {za: a E A} E Zo, then g(z) = {4,0(x0): a E A). Clearly, g is a continuous map from the compactum go onto the compactum Z. Since 00(0) = 0 for all a E A, we have g(Z0) C Z. We claim that g-'(7,) C Z0. In fact, if a point z = {za: a E Al has uncountably many nonzero coordinates, then 4-'(0) = {0} implies that the point g(z) = {4,0(z0): a E Al has uncountably many nonzero coordinates. Hence, if z 4 Zo, then g(z) Z, i.e. g`(Z) C Z0. This implies that the compactum g-'(B) is contained in Zo. However, as noted above, Z0 is homeomorphic to a subspace of Cp(LT, D). Consequently, the compactum X = g' (B)

is homeomorphic to a compactum in Cp(LT,D). Note that g(X) _ g(g-'(B)) = B, since g(Z0) =7Necessity in theorem 3.1. Let B be a Corson compactum of weight < r. By proposition 3.9, the compacturn B is the image of a compactum X in Cp(LT,D) under the continuous neap g. The map g: X B is closed, since X is a compactum. Hence Cp(B) is homeomorphic to a closed subspace of Cp(X ). By assertion 3.8, Cp(X )

is a primary Lindelof space. But then so is CC(B). This proves the necessity part in theorem 3.1. A key role in the the proof of the sufficiency part in theorem 3.1 is played by the following proposition, due to S. P. Gul'ko.

IV.3.10. Proposition. If X is a primary Lindelof space, then Cp(X) can be mapped by a one-to-one continuous linear map into the E-product of a certain number of copies of the real line.

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Proof. If Y is a continuous image of X, then CC(Y) is linearly homeomorphic to a subspace of CC(X) (chapt. 0). Therefore it suffices to prove proposition 3.10 for the case when X is a closed subspace of a space (L,)", where r > Lio. Fix such an X. We agree on some notation. Let A C L,. By RA: (L,)" -. (L,)" we denote the map assigning to an arbitrary point x = {xi: i E N+} the point y = {yi: i E N+} for which yi = xi if xi E A and yi = t; if xi A. The map RA is clearly a continuous retraction of (L,)" onto a subspace of it. Let n E N+, V a neighborhood of 6 in L and t = (z1,... , zn) E (L,)". If zi

we put Vi = {zi}. If zi = 1, we put V = V. By it, V] we denote the open set in (L,)". If f o r t = (z1, ... , zn) there is a neighborhood V1x x V x L, x L, x V of £ in L, such that It, V] n X = 0, then some such V is denoted by V (t), and t is called a distinguished point. Put C(t) = L, \ V(t), and note that C(t) is countable.

IV.3.11. Lemma.

Let X be an arbitrary closed set in (L,)". Then for any set

A C L, there is a set M* (: L, such that A C M*, I M'I= JAI- No, and R5,T. (X) C X.

Proof of lemma 3.11. The set M* is constructed by induction. Put MI = A U {c}. Suppose that Mn C L, has already been constructed, in such a way that IMnI < JAI No. Denote by Tn the set of all distinguished points t = (y1, .. , , yk), where k < n and y 1 , . . . , yk E Mn. Clearly, JTnJ < JAI No. Put Mn+1 = Mn U U{C(t): t E Tn}. Clearly, JMn+11 < IMni 1 o < JAI

No.

We show that M' = U{Mi: i E N+} is the set looked for. We have A = M1 C M* and IM'I < JAI Lto. It remains to verify that RM. (X) C X. Take an arbitrary point y = (y1, ... , yk,...) E (L,)" for which RM. (y) 0 X, and put z = RM. (y), z = (z1, Z2.... ). There is a standard basic neighborhood of z not intersecting X. Hence we can choose a neighborhood V of C in L, and an index n E N+ for which It, V] n X = 0, where t = (z1i... , zn) E (L,)n. By the definition of RM., we have z1,.. . , zn E M* = U{Mi: i E N+}. Since Mi C M;+1 for i E N+, there is an n' E N+ such that n < n' and z1,... , zn E Ma.. Then t E Tn., and hence there is defined a neighborhood V (t) of e for which It, V (t)] n X = 0. The definition of the set Mn.+1 implies that C(t) = L, \ V(t) C M'. We show that y E [t,V(t)]. Take an arbitrary i < n. If zi L , then V(T) = {zi}, and zi E M* \ f}. Then yi = zi, i.e. yi E V (t). Let zi = C. Then Vi(t) = V (t), and either yi = C or yi 0 M*. In the first situation, Vi E V(t) = ,(t), and in the second yi 0 C(t) = L, \ V(t), since C(t) C M', and hence yi E V(t) = V (t). So yi E Vi(t) for all i = 1, ... , n, i.e. V E [t, V(t)]. Since It, V (t)] n X = 0, we find that y 0 X. Lemma 3.11 has been proved. Below we denote by cf(r) the smallest cardinal A such that the cardinal r can be represented as a sum of at most A cardinals smaller than r.

IV.3.12. Lemma. Let A = cf(r), and let X be a closed set in (L,)". Then there is a transfinite sequence {Ma: a < A} of subsets of L, such that the following conditions hold:

IMaJ
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14L

2) L, = U{Ma: a < Al; 3) Ma = U{MO: /3 < a} for every limit ordinal a < A; 4) Rh4, (X) C X for all a A.

Proof. The set L.,. can be represented in the form L, = U{Ba: a < Al, where IBaI
< a}. We now show by transfinite induction that if X is a closed subspace of (LT)", then C(X) can be linearly condensed into a E-product, of copies of the real line. Assume that proposition 3.10 is true for all cardinals less than T. Let A = cf(r). Take a family {Ma: a < Al as in lemma 3.12, and put Ra = RM,, Xa = Ra(X ). The correspondence f -f f o (RaI x) is a linear homeomorphism from Cp(,Ya) onto the subspace Ta = {f o (Ralx): f E C,(X)} of CC(X). Clearly, Ya = X fl (Ma)" and, since IMaI < T, by the induction assumption there exists for each a < A a linear condensation ¢a from Ta onto a subspace of the Eproduct E(I'a) of ra copies of the real line R, where Iral < T. Then we glue together the maps ci by a formula due to Amir and Lindenstrauss: for each f E CC(X), 5) q5(f) = {¢a+r (f o (Ra+l lx) - f o (Ral x)): a < Al, where q5(f) is regarded as an element of the Tikhonov product P = 11{E(ra): a < Al. Clearly, for every a < A the map from CC(X) into E(ra+I) given by the formula Yea+l(f o (Ra+rlx) - f o (RaI x)) is continuous. Hence the map 0 that is the diagonal product of these maps is continuous too. We verify that under 0 distinct points have distinct images. First we note that the set Y,:3 = U{Xa: a < 0} is everywhere dense in XX, for every limit ordinal /3 < A (where Y,, = X). In fact, if x E XX, x = (xI, ... , xk, ... ), then there is for each n E N+ an an < A for which XI,. .. , xn E Ma,, by properties 1)-3) of the family {Ma: a < Al. Then every standard neighborhood of x with base x1, ... , xn contains the point (x I .... , x,,, e, C, ... ), which belongs to Xa,, = Ra, (X) Let now f,g E CC(X) and f g. Since Y,, is everywhere dense in X, there is an

ordinal a < A for which fix, - glx,,. Let ,6 be the first such ordinal a. Then /3 is not a limit ordinal, since otherwise Yo = a' < ¢} would be dense in Xf and we could find an a' < /3 with f IX., glx,,, contradicting the choice of /3. Hence /3 = a'+1 for some a' < A. By the choice of /3 we have f o(Ra,+rlx) go(Ra'+l Ix) and

fo(Ra'Ix)=go(R, ix). Thus, fo(Rar+llx)-fo(Rarlx)

g°(Ra"+rlx)-go(Ra,lx)

Since Oa+I maps distinct points to distinct images, we conclude that 45(f) 54 ¢(g).

It remains to verify that for each f E Cc(X) the point 45(f) in the product P = jj{E(I'a): a < Al has at most countably many nonzero coordinates.

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In other words, we must prove that the set S = {a < r: f o(R°4.i1Y)

f o(R°l.r)}

is countable.

Assume this to be not true. Choose for each a E S a point x° E X such that 6) foR°+1(x°) f o R.(xa). Since S is uncountable, there are an uncountable set S' C S and a number e > 0 such that 7)

I f o R°+1(x°) -foR°(x°) > e for all a E S'.

The space X is Lindelof, being a closed subspace of the Lindelof space (L,)'. Since {R°(x°): a E S'} C X and S' is uncountable, there is a point z = (z1, ... )zk, ...) E X for any neighborhood W of which the set {a E S': &(x°) E W} is uncountable. The function f is continuous at z. Therefore there is a neighborhood U of z of the x V. x L, x L, x for which the diameter of the set f (U) is less form U = VI x than e. We may also assume that if i < n and zi 0 , then V = {zi}, while if z1 = E and i < n, then Vi = V, where V is a neighborhood of in L, not depending on i. By the choice of z, the set S" = {a E S': R°(x°) E U) is uncountable. Since the family {M°+1 \ M°: a E S"} is disjoint and consists of nonempty sets, while the set L, \ V is countable, there is a Q E S" for which Ms}1 \ AM1s C V. Moreover, Q E S" implies that Ro(xo) E U. We verify that also Rs+1(xs) E U. We first note that the definitions of Ro+l and Rs immediately imply the following assertion: 0) if for some i E w the ith coordinate of Ro+1(xs) belongs to the set Mo, then this coordinate coincides with the ith coordinate of the point Rs(xo). Put now y' = Ro(xs), x' = Rs+1(xe), and compare the ith coordinates of the points

x', y', and z for i
Now y' E Ro(Xo)

implies that x = y; E Ms C Ms+1 This implies that x; = yi = zi E V. Let i < n and zi Then V = V and, since y' E U, we have y; E Vi. By the definition of Ro+1i xi E Mo+I Two cases are possible. If x; E Ms, then, by 0), x; = y; and hence x; E V,1. If x; 0 Ms, then x; E Me+I \ Ms C V = V by the choice of Q. In any case,

x;EVi for all i3 E S. Proposition 3.11 has been proved.

Sufficiency part in theorem 3.1. Let X be a compactum, and let C,(X) be a primary Lindelof space. By proposition 3.10, the space CPCP(X) can be linearly condensed onto a subspace of the E-product ER' of irk copies of the real line, for some set r. But X is homeomorphic to a subspace of CPC,,(X) (see chapt. 0). Since a condensation on a compactum is a homeomorphism, we conclude that the compactum X is homeomorphic to a subspace of ER". Theorem 3.1 is completely proved. Theorem 3.1 and proposition 3.10 imply

IV.3.13. Corollary. If X is a Corson compactum, then the space CCC(X) can be linearly condensed onto a subspace of the E-product ER" of in copies of the real line,

for some set r.

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In fact. as has been proved by S. P. Gul'ko, if X is a Corson compactum. then all spaces Cp(X ), CCC(X),... , C,, C,,(X) allow a linear condensation onto a subspace of the E-product of copies of the real line. This implies that if X is a Corson compactum, then every compactum in a space Cp,,,(X) for some n E N+ is a Corson t' compactum. Using this, it can be readily proved that if X Y for some n. E N+, where X and Y are compacts and X is a Corson compactum, then Y is also a Corson compactum. Theorem 3.1 allows us to obtain the following, much weaker, result.

IV.3.14. Corollary.

If X t Y, where X and Y are compacta and X is a Corson

compactum, then Y is also a Corson compactum.

The following result is due to S. P. Gul'ko, E. Michael, and M. E. Rudin.

IV.3.15. Corollary.

The image of a Corson compactum under a continuous map

is a Corson compactum.

Proof. Let f : X

Y be a continuous map from a Corson compactum X onto a compactum Y. Then C,(Y) is homeomorphic to a closed subspace of Cc(X). By theorem 3.1, Cc(X) is a primary Lindelof space. Hence C,,(Y) is a primary Lindelof space. Applying theorem 3.1 again, we conclude that Y is a Corson compacturn. A space that can be represented as a continuous image of a closed subspace of the product of the space (LT)' and a compactum is called a k-primary Lindelof space. In relation with theorem 3.1 there naturally arises the following

IV.3.16. Problem.

Let X be a compactum and CP(X) a k-primary Lindelof space. Is then X a Corson compactum? (Recently, I. Bandloff notified me that he has answered this question positively.) Proposition 3.10 implies

IV.3.17. Corollary.

If X is a primary Lindelof space, then every compact urn in Cp(X) is a Corson compactum. Corollary 3.17 implies

IV.3.18. Corollary. If X is a compactum and CC(X) contains a primary Lindelof space separating the points of X, then X is a Corson compactum.

This generalizes somewhat the sufficiency part of theorem 3.1. Note that, as has been proved by A. G. Leiderman [31[, assertion 3.17 cannot be generalized to Lindelof

P-spaces: there are a Lindelof P-space X and a compactum F C Cp(X) such that F is not a Corson compactum.

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4. Resoluble compacta, and condensations of CC(X) into a E.-product of real lines. Two characterizations of Eberlein compacta The scheme of reasonings given in this section is due to S. P. Gul'ko (261. In particular, we will characterize Eberlein compacta as special Corson compacta, embedding

them in a certain part of a E-product of real lines, called the E.-product. We will also give an intrinsic description of Eberlein compacta in the language of bases. We agree on the following notation. Below, X denotes a space, M a subspace of

it, Y a subspace of the space CP(X), and L a subspace of Y. By 7rM and Oy we denote, as usual, the canonical restriction map and the canonical evaluation map, 7rM: C,(X) -. CC(M), 1Iy: X -+ CC(Y). Below, z(iy is also called the reflection map with respect to Y. We also consider the evaluation map 4'L: X -+ CC(L).

IV.4.1. Proposition. Suppose Y separates the points of X, M is compact, lrM(L) is everywhere dense in lrM(Y), and OL(M) is everywhere dense in i/it(X). Then zbt(M) = OL(X), the map 1GLIM: M -+ biL(M) is a homeomorphism, and there is a continuous retraction pM : X Al, defined by PM = (IPLI M)_I - 'OL

(1)

Moreover, for every x E X and every f E L the relations 'bL(PM(x)) = V'L(x)

(2)

f(x) = f(pM(x))

(3)

and

hold.

Proof. The map 'PL is continuous, therefore /L(M) is a compactum, and hence closed in 1GL(X). Consequently, 't(M) = tiL(X). The map VILI M: M - VIL(M) is a condensation (see chapt. 0). Since M is compact, we conclude that 1I'LIM is a homeomorphism, and hence the inverse map (?'LIM)-1: T/iL(M) = t/iL(X) -+ M is continuous. Clearly, pM(x) = (VILIM)-1 01PL(x) = x for all

tEM.

Thus, pM: X -+ M is a continuous retraction. Formula (2) clearly, holds: W

L(PM(x)) = 1'L ° ('LI M)-1 0Ybj r) = ?PL (x)

Relation (3) is merely another way of writing (2).

We say that the sets M and L are conjugate (with respect to the pair X, Y) if lrM(Y) = 7rM(L) and biL(M) = tPL(X).

IV.4.2. Lemma. Let A C X, B C Y, and suppose that lrA(B) is everywhere dense in lrA(Y) and 1I)B(A) is everywhere dense in 1GB(X).

If A and B are compact,and B C Y, then A and B are conjugate.

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145

Proof. Since xA(B) and l1B(7) are compacta which are everywhere dense in, respectively, lrA(Y) and OB(X), we have nA(B) = irA(Y) and rbB(A) ='B(X). If it (a) = l'B(x), then iiy(a)IB = t y(x)IB, and since the functions 1l',-(a) and ipy(x) are continuous on Y, we conclude: z/1y(a)Ig = '(x)( , i.e. ta(a) = B(x). (X). Thus, OB(A) = OB(X) implies that ipp-(A) _ Also, lrA(B) = 7rA(Y) implies that ir-(B) _ irA(Y), since the values of a continuous function are uniquely determined by its values on an everywhere dense set.

IV.4.3. Lemma.

Let X and Y be compacta, r) a limit ordinal, and suppose that for all l; < 17 there are defined sets At C X and BB C Y such that: a) At C At, and Bf C Bt' for £ < £' < 17; b) 7rAE(Bt+1) is everywhere dense in rAe(Y); c) xbbBE (At+1) is everywhere dense in 'PBE (X ).

< ij} and L = U{BF

Then the sets M = U{A( :

< 17} are conjugate..

Proof. By b), 1rAE(L) J 1rAE(Bt+I) implies that 7rAf(L) is everywhere dense in irAE(Y)

for all t; < rt. Hence, using the definition of topology of pointwise convergence and condition a), we conclude that 7i-AI(L) is everywhere dense in 2rM(Y). Similarly, conditions a) and c) imply that is everywhere dense in 1'L(X)-

Bur M and L are compacta, L C Y. Lemma 4.2 now implies that M and L are conjugate. The following particular case of lemma 4.3 is important in the sequel.

IV.4.4. Proposition.

Let X and Y be compacta, rl a limit ordinal, and suppose that for all < q there are defined sets At C X and BB C Y such that:

a) AtCAt, andBBCBB' forC
Then the sets M = U{At :

t:

17} and L = U{Bt

< rl} are conjugate.

IV.4.5. Lemma. Let X and Y be compacta, r a cardinal, r > Ho, AI C X, B1 C Y, and IAII < r, IBII < r. Then there are sets A C X, B C Y such that A 1 c A, BI C B, IAI < r, IBI < r, with M = A, L = B conjugate sets. Proof. By induction we can in an obvious manner construct sets An C X, Bn C Y, for n E N+, subject to the following conditions: a)

IA8I

IB8I
b) An C An+1, Bn C B,+i; c) lAn (Bn+1) everywhere dense in 7rA (Y); d) ?PE (An+1) everywhere dense in 0Bn (X ).

Fulfilling these requirements is not difficult since the weights of the spaces 1rAn(Y) and OBn(X) do not exceed the cardinalities of the sets An and Bn, respectively, i.e. are not larger than T.

146

IV. I.INDELOF NUMBER TYPE PROPERTIES FOR FUNCTION SPACES

PutA=U{An: nEN+},B=U{Bn: nEN+}. Then JAI
IV.4.6. Proposition.

Suppose we are given, for all n E N+, sets Mn C X and

Ln C Y, satisfying the following conditions: a) Mn C Mn+l , Ln (Z Ln+1; b) Mn and Ln are conjugate. Let, moreover, X and Y be compacta and Y separate the points of X. Then for any sequence {xn: n E N+} of points in M = U{1Lin : n E N+} converging to a point x E X, the sequence {pn(xn): n E N+} also converges to x, where pn: X Mn is the retraction corresponding to the conjugate pair Mn, L.

Proof. Take an arbitrary m E N+ and an arbitrary f E L,n. Let n E N+ and n > m. Then f E LM C Ln and f (xn) = f (pn(xn)) by formula (3). x,, = x, we have limn....,. f (xn) = f (x). Thus, also Since f is continuous and limn f (pn(xn)) = f W. On the other hand, there is a point y E M that is a limit point for the sequence {pn(xn): n E N+}, since M is compact. But limn f (pn(xn)) = f W. Thus f (x) = f (y). Using the fact that m E N+ and f E L,,, are arbitrarily chosen, we conclude that f (x) = f (y) for all f E U{Ln: n E N+}. But then also f (x) = f (y) for all f E L = U{Ln : n E N+}, by the definition of topology of pointwise convergence.

We assume that x # y, and derive a contradiction from this. There is a g E Y such that g(x) 0 g(y), since Y separates the points of X. But lrM(L) is everywhere dense in 7rM(Y) (for M and L are conjugate by proposition 4.4). The set M is closed, hence x, y E M. This implies that f (x) # f (y) for some f E L. We have obtained a contradiction. Hence x = y. We have shown that x is a limit point for the sequence {pn(xn): n E N+} and, moreover, the only such point in X. Since X is compact, we conclude that the sequence {pn(xn): n E N+} converges to x.

IV.4. '. Proposition.

Let X be a compactum of density T, and Y a compactum separating the points of X. We can then determine, for each [; < T, sets Me c X and Le C Y such that the following conditions are satisfied:

1) McCMe' and LLCL(, fort <[;'
4. RESOLUBLE

AND CONDENSATIONS OF Cp(X)

147

3) the density of ME does not exceed 120.11, and the density of L( does not exceed

4) if r) < r and 77 is a limit ordinal, then

M,,=U{M :
Proof. Fix an everywhere dense set Q = {X.: a < r} in X. The sets Mt and Lf are defined by transfinite recursion. We may take M0 = 0 and L0 == 0. Let n < r, and suppose that for all < 77 sets Aft and Le have been defined, satisfying conditions 1)-3) and the additional condition: 7) .ra E M£ for a < C. We first assume that. rl is a limit, ordinal. We then put. M,, = U{M( : <,l} and L,, = U{LF : < rl}. By proposition ,1.4, the sets Al,, and L. are conjugate. Clearly, 1), 3), 4), and 6) are also fulfilled. Moreover, condition 7) is fulfilled for = rl::r, E M,, if a < 11. Let rl be an isolated ordinal, rl = o + 1. By 3) we can take everywhere dense sets A0 C M&, and Bo C L&, in Alto and Lto, respectively, such that JA0J < lCol, JBol < ICol.

Put Al = A0 U {x0: a < eo} and B, = Bo. By lemma 4.5 there are sets A C X and

BCYforwhich ADA,,BDB1,IAA=JA,
3') U{pf:
< rl};

5') if a sequence {x,,: n E N+} C X converges in X to a point x E X and a n e N+) D sequence {fin: n E N+} C r is such that Cn < Cn+, and {xn: n E N+}, then the sequence {N. (xn): n E N+} also converges to x. A compactum is called resoluble if there is on each subcompactum in it a resoluble family of retractions. Clearly, every hereditarily separable (over closed sets) compactum is resoluble. In particular, every metrizable compactum is resoluble.

IV.4.8. Theorem. Every Eberlein compactum is resoluble.

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IV. LINDELOF NUMBER TYPE PROPERTIES FOR FUNCTION SPACES

Proof. Let X be an Eberlein compactum of density r. We show that there is a resoluble family of retractions on X. There is a compactum Y C C,,(X) separating the points of X. By proposition 4.7 we can choose families {Mf: C < r} and {Le: < r} of sets in X and Y, respectively, satisfying the conditions 1)-6) in proposition 4.7. To the pair Mt, Lf there uniquely corresponds a retraction pe : X - M(, generated by it (see proposition 4.1). The family of retractions {p{: < r} is the family looked for. In fact, compare the conditions 1')-5') with the conditions 1)-5) in proposition 4.7. Clearly, 1') follows from 3) (since pf(X) = M£ and IfI < r). 1) implies 2'), and 5) implies 3'). The corresponding condition 4') is just a somewhat rewritten version of 5).

Finally, 5') follows immediately from proposition 4.6 and conditions 1) and 2). We have constructed a resoluble family of retractions on the entire Eberlein compactum X. However, every compactum in X is itself an Eberlein compactum. Thus every Eberlein compactum is resoluble. Theorem 4.8 has been proved.

Let X be a Frechet-Urysohn compactum, r an infinite cardinal, and suppose we are given a family {pf: C < r} of continuous retractions of X. For f < r we put Mf = pf(X), and consider the maps pE: C(X) - C(X), irf: C(X) , C(M(), and of : C(X) -+ C(X) defined by pf(f)(x) = f (p( (X)) for all x E X;

(4)

7rt(f) = fIMF; (5) (6) gf(f)(x) = f (x) - f (pf(x)) for all x E X. Clearly, the maps pt, ire, and qe are continuous linear maps with respect to the topology of pointwise convergence on C(X) and C(MM). Hence the map he = 7rt+I o of : CP(X) - Cp(Mf+I)

is also continuous and linear. It is convenient to denote by h_1 the restriction map fro: Cp(X)

(7)

Cp(Mo) (where

0 is the smallest ordinal), and to mean by c> < r that -1 < c < r. With these notations we have the fundamental

IV.4.9. Proposition.

Let X be a Frechet-Urysohn compactum, and let a family {pf: £ < r} of continuous retractions of it satisfy the conditions 2')-5') in the definition of a resoluble family of retractions. Now: I) if fl, f2 E C(X) and f1 f2i then there is a to such that ht(f1) 0 hf(f2);

II) for any f E C(X) and any f > 0 the set

< r: there is an x E Me}1 such

that Ihf(f)(x)I > e} is finite. Proof. I. Since U{Mf : C < r} = X, there is a L <,r such that fi l ME f2IM4 Denote by 971 the smallest element in the set {C < r: f1IME f2lmt}. By condition 4'), 171 cannot be a limit ordinal. Thus, either 171 = rto + 1 or rlt = 0. In the second case h_1(f1) = film. 0 f2IM0 = h_1(f2), i.e. we find C = -1.

4. RESOLUBLE COMPACTA, AND CONDENSATIONS OF Cp(X)

149

Let t71 = 71o + 1. Take a y E M,}1 for which fl(y) # f2(y). By the choice of 4,ro(fi)(y) = fl(y) film,,,, = f2Jm,,. Thus, fi(p, (y)) = f2(pgo(y)), whence fl(p,.(y)) V- f2(y) - f2(p,,.(y)) = 4.ro(f2)(y). Since y E M,ro+1, we conclude that

711,

4,,j(fl)IM,,a+, J 9go(f2)In.j,,o+ i.e. h,ro(fl) # h,(f2), and rlp is the value t; looked for.

11. Assume the contrary. For some f E C(X) and some e > 0 there are then a sequence {Cn: n E N+} C r and a sequence {x,,: n E N+} C X such that Cn, # fn,, if

WO n", and Ihtn(f)(xn)I> eforallnEN+. Since r is a well ordered set, and X is a Frechet-Urysohn compactum, without loss of generality we may assume that the sequence {xn: n E N+} converges to a

point x E X, and that Sn < ,,+1 for all n E N+; if not, we pass to appropriate subsequences.

We have X. E M (.+j C M{ + since cn + 1 < (n+l. Thus, {xn: n E N+} C U{ME,,: n E N+} n E N+}. Using condition 5'), we now conclude that the sequence also converges to x. Hence limp AR. (xn)) = f (x). Therefore there is an m E N+ for which I f (xm) - f (pP,n (xm)) I < e, i.e. I hem (f) (x,n)I < e, contradicting the choice of j,.

We are now ready to prove the main result. It is only necessary to recall the definition of E.-product of real lines. By R, Ra, or Rn we denote the real line with the ordinary topology. The subspace of the Tikhonov product jj{Ra: a E A) formed by those x = (xa: a E A) for which

the sets {a E A: Ixal > e} are finite for all e > 0 is denoted by E.{Ra: a E A} or E.(A), and is called a E.-product of real lines (with respect to the index set A). For X E E.(A), x = (xa: a E A), we let, as usual, Ilxll = max{lxal: a E A}.

IV.4.10. Theorem. Let X be a resoluble Frechet-Urysohn compactum. Then the space C,,(X) can be linearly condensed onto some (not necessarily closed) subspace of a E A} (where JAI = d(X)); moreover, without the linear topological space

increase of the norms (i.e. in such a way that if f E C(X) is mapped to z E E.(A), then IJzJJ < llfll)

The case of a separable X is covered by the following lemma.

IV.4.11. Lemma.

If X is a separable compactum, then CC(X) can be linearly condensed into E.(N+), even without increase of the norms.

Proof. Take a countable everywhere dense set {xn: n E N+} in X. Consider the map 0: Cp(X) -+ lI{Rp: n E N+} defined by (rn : n E N+), where rn = n f (xp).

(8)

Clearly, ¢ is continuous, linear, and ¢(fl) 4'(f2) if fl f2. Since X is a compactum, every function f E C(X) is bounded. Thus, 4,(C(X)) C E. (N+), and 0 is the required condensation.

Proof of theorem 4.10. We reason by transfinite induction with respect to the density d(X) of X. For d(X) < No theorem 4.10 has already been proved. Suppose that the

IV. LINDELOF NUMBER TYPE PROPERTIES FOR FUNCTION SPACES

150

assertion is true when d(X) < r, with r some cardinal, T > 1 o. We show that the conclusion of theorem 4.10 is true also when d(X) = r. Since X is a resoluble compactum, there is resoluble family of retractions {pf:


(9)

Since r > 1Zo, the density of the compactum M4+1 is less than r (see condition 1')).. But. Mf+1 is a resoluble compactum (for every compactum in a resoluble compactum

is resoluble). Thus, by the induction hypothesis, the conclusion of theorem 4.10 is valid for the space C1,(MM+1) for -1 < C < r: there is a linear condensation gg, which is not norm increasing, from Cp(Mf.1.1) onto some linear subspace of a space E.{R,,: a E Af}, where IA41 = d(Mf}1) < r. Without loss of generality we may assume that

Af.f1A,,,=0 forte' i;". Let hf = hf/2, i.e. hf (f) = hf (f)12 for f E Cp(X ). Clearly, the map hf 'distinguishes' the same points of C(X) as hf does, and, moreover, does not increase norms, as is clear from (9). Finally, let gf denote the composite of the maps gf and h.f:

gf=hfogf:Cp(X)-.E.{R.: aEAt}; it is a continuous linear map which is not norm increasing. Since {pf: < r} is a resoluble family of retractions on X, for two arbitrary f1, f2 E C(X) there is a C < r, C > -1, such that hf(f1) hf(f2) (by proposition 4.9.1). Since gf is a condensation, also g4(f1) 0 9f(f2). Consequently, the diagonal product

g = i{gf : -1 < 6:5,r}: Cp(X) -. 11{E.(Af) : -1 <% < T} is a linear condensation 'into' which is not norm increasing. Put A = U{Af: -1 < C < T}. Clearly IAI < r. We have g(Cp(X)) C Fj{E.(Af): -1 < e < T} C j1{R,,: a E A}, and proposition 4.9 and the fact that every gf is not norm increasing imply that g(Cp(X)) belongs to E.{Ra: a E A}. Theorem 4.10 has been proved.

We now give two very important characterizations of Eberlein compacta. The first is due to Amir and Lindenstrauss [70], and the second to Rosenthal [141]. These characterizations are strongly related, but there is also a principal distinction between them: one of them is in terms of embedding, while the other has a purely 'intrinsic' character.

IV.4.12. Theorem [70]. A compactum is an Eberlein compactum if and only if it can be homeomorphically mapped into the E. (A) product of real lines, for some A.

4. RESOLUBLE COMPACT., AND CONDENSATIONS OF Cp(X)

151

Proof. Sufficiency. Let F be a compactum homeomorphic to a subspace of E.(A). We compactify the discrete set A by adjoining to it a new point z; whose neighborhoods

are the complements in A U {l;} of finite subsets of A. The compactum obtained is the so-called Aleksandrov supersequence A where r = JAS. Clearly, the space E.(A) is canonically homeomorphic to the subspace If E C,,(A,): f ((:) = 0}. Hence we may assume that F C C9(A,). Since A, is compact, we conclude that F is an Eberlein compactum. Necessity. Let F be an Eberlein compactum. By IV. 1.6, we may assume that

F C C9(X), where X is a compactum, and also that F separates the points of X. Then X is an Eberlein compactum, and hence (theorem 4,8) a resoluble compactum. By theorem 4.10, Cc(X) can be condensed into E.(A), where the cardinality of A equals the density of X. Moreover, the compactum F C CG(X) is homeomorphically mapped into E.(A).

IV.4.13. Corollary.

For an arbitrary cornpacturn X the following conditions are

equiva.lcnt:

a) X is an Eberlein compactum; b) Cp(X) contains a compactum F separating the points of X which is horneomorphic to some Aleksandrov supersequence A,; c) X is homeomorphic to a compactum in Cp(A,), for some r.

Proof a) implies c) by theorem 4.12 and the fact that, E.(A) is homeomorphic to a subspace of CD(A,), where r = JAS (see the proof of theorem 4.12). c)b). If X C CP(A,) , then the image of the compactum A, under the canonical map 1,hx: A, --+ Cc(X) is clearly a compactum with at most one nonisolated point, and is hence isomorphic to A,, for some r'. The compactum zlix(A,) = A,, separates the points of X. b) implies a) by proposition IV.1.6. As usual, a cozero set in a space X is any set of the form {x E X: f (x) 0 0}, where f is a realvalued continuous function on X. A family of sets is called o-point-finite if it is the union of a countable family of point-finite sets. Finally, a family -y of subsets of a space X is said to be To-separating (the points of X) if for two arbitrary distinct points x, y in X there is a V E y containing precisely one of these points. Note that in compacta the cozero sets coincide with the open F, -sets.

IV.4.14. Theorem (141). A compactum X is an Eberlein compactum if and only if it contains a To-separating a-point-finite family of cozero sets.

Proof. Necessity. Let X be an Eberlein compactum. By theorem 4.12 we may assume that X C E. (A) for some A. We denote by Q* the set of all rational points distinct from zero. Let Q E A. For

rEQ`andr>0weputV,,p={x=(x,,: aEA): xp>r). ForrEQ`andr<0

we put V,,a = {x = (xe: a E A): xx < r}. It is obvious that V,,5 is also in this case an open Fe-set, and even a cozero set, in E. (A). Put ry, = {V,,,,: a E A), for r E Q.

IV. LINDELOF NUMBER TYPE PROPERTIES FOR FUNCTION SPACES

152

Each point x = (xa: a E A) of E.(A) belongs to only finitely many elements of the family ryr for r E Q` fixed, since otherwise the set of coordinates of this point which exceed in absolute value the number Irl > 0 would be infinite, contradicting the definition of E.(A). Thus, ryT is a point-finite family, and £ = U{-j,.: r E Q'} is a o-point-finite family of open F, -sets in E.(A). Note that, although we will not need this fact, only the point with all coordinates zero is not covered by E. We show that £ To-separates the points of E. (A). Let x, y E E. (A), x # y. There

is an aEAsuch that xa

ya. Take an rEQ*for which xa
Clearly, V,,, then contains precisely one of the points x, y. The trace of £ on X (i.e. the family £o = {VnX: V E 6}) is clearly a To-separating o-point-finite family of open F, sets in X. Since X is compact, all U E Eo are cozero sets. Sufficiency. Suppose we are given, for n e N+, families ryn = {U,,n: a E An} of cozero sets in X such that £ = U{7'n: n E N+} is To-separating the points of X, and every ryn is point finite. Without loss of generality we may assume that for distinct n the sets An are disjoint; this allows us to write Ua instead of Ua,n. For each n E N+ and each a E An we fix a function f, E C(X) such that 1h fa it < l /n and X \ Ua = {x E X: fa(x) = 0}, i.e. X \ Ua is the zero set of the function f"'. Put A = U{An: n E N+}. Note that for each a E A there is a uniquely defined function

fa,since An,nAn,,=0ifn'0n". The map Vi: X -* E.(A) is defined by ?P(x) = y = (y,,,: a E A), where ya = fa(x)

for all aEA. The map 0 is continuous, since all fa are continuous. If x', x" E X, x' x", then there is a 8 E A such that Up contains precisely one of the points x', x". Suppose, e.g., that x' E Up and x" 0 Up. Then fp(x") = 0 and fp(x') 0 0 by the definition of fp. Thus, yp(x') yp(x") and 0(x') 0 ii(x"). But X is a compactum and E.(A) is a Hausdorff space. Hence V homeomorphically maps X onto some compactum in E.(A), and by theorem 4.12 X is an Eberlein compactum. Theorem 4.14 has been proved.

By making some obvious changes in the proof of theorem 4.14 we arrive at the following result.

IV.4.15. Theorem.

A compactum X is a Corson compactum if and only if it

contains a point-countable To-separating family of open F,-sets.

5. The Preiss-Simon theorem The main theorem in this section, theorem 5.1, relies heavily on theorem 4.12. It deepens our understanding of the peculiarities of limit transition in Eberlein compacta. Below we present, with minor changes, the reasoning in [137], in which this theorem appeared for the first time.

A sequence {An: n E N} of subsets of a space X converges to a point x E X if every neighborhood of x contains all An from some sufficiently large n onwards.

A space X is called a Preiss-Simon space if for each closed subset Y of it and each point y E Y there is a sequence {Un: n E N} of nonempty open sets Un in Y,

5. THE PREISS-SIMON THEOREM

153

(Jr, C Y, converging to y.

Clearly, the adjective 'closed' can be dropped in this definition, without changing

the scope of the notion of Preiss-Simon space. Every Preiss-Simon space is a Frechet-Urysohn space; this is obvious. In this section we prove that the converse is false. Compact Preiss-Simon spaces are also called Preiss-Simon compacta.

IV.5.1. Theorem [137].

Every Eberlein compactum is a Preiss-Simon com-

pactum.

Proof. Let X be an Eberlein compactum. Since every closed Y C X is again an Eberlein compactum, it suffices to consider the case Y = X. Let y E Y = X. By theorem 4.12 and the fact that E,(A) is a topological group, we may assume that X C E. (A) and y,, = 0 for all a E A. The following fact plays a crucial role: *) for each (open) neighborhood U of y and each e > 0 we can find a finite set.

13' c A and a nonempty open set V in X, V C U, such that, Ixal < e for all

aEA\B'andallx=(x,,: aEA)EV. We prove *).

Fix a neighborhood U of y. We take U to be open in X. For each B C A we put

KB={xEX: Ixal>efor all aEB}, and KB=Ix EX: Ixal>eforallaEB}. Clearly, KB C KB, and I(B is a closed set in X. Since X C E.(A), the sets KB and KB are empty if B is infinite. Thus, KB is open in X for any B C A. Consider the family P of all sets B C A for which KB fl.U # 0. If B E P, then the closed set KB cannot be empty. Clearly, every B E P is finite.

LetB=U{Bi: ieN+}, where B,CBi+1 and B,EPforalliEN+. It is then

obvious that KB:+1 C KB: and 1(B = fl{KB;: i E N+}. But KB, is a nonempty closed

set in the compactum X, since Bi E P. Thus, KB # 0, and hence B is finite. Thus, we have proved that P does not contain infinite strictly increasing sequences of sets. Hence P contains a maximal (with respect to inclusion) element. Let B' E P be such.

Then V = KB, fl U is a nonempty open set in X (since B' E P), V C U, and the maximality of B' in P implies that IxaI < e for all x E V and all a E A \ R. Condition *) has been verified. We define by induction for each n E N+ a neighborhood U,, of y in X, an open set If,, in X, and a finite set Bn C A, as follows.

We take U1 = X and B1 = 0. Let n E N+, n > 1, and let the finite set Bn_1 C A and the open set Un_1 in X containing y already been defined. Put Un = {x E U._1: IxaI < 1/n for all a E Bn_1}. Clearly, y E Un and Un is open in X. By *), for e = 1/n we can choose a nonempty open set Vn in X and a finite set B;, C A such that Vn C Un = {x E Un_1: Ixa) < 1/n

if aEBn_1},and IxaI <1/n for all aCA\B,,and all x Vn. Take Bn to be the finite set Bn_1 U B,', C A.

The sets Un, Vn, and Bn are now defined, and it is also clear that the conditions Vn+1 C Un+1 C Un and Bn C B,,+1 hold.

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IV. LINDELOF NUMBER TYPE PROPERTIES FOR FUNCTION SPACES

Consider an arbitrary finite set C C A and an e > 0. There is clearly an n E N+ for which 1/n < e, n > 1, and Cfl Bn = Cfl (U{B;: i E N+}). For any x E Vm, where m > n + 1, we then have:

a) if aECand aEBn,then aEBm_land Ix.Ql <1/in
Vrn C Urn (see

the definition of Um);

b) if a E C and a V Bn, then a V Bm and, moreover, a. B;,,, whence, by the choice of Bm and Vm, Ixal < 1/m < e.

Thus, Vm C {x E X: Ixaj < e for all a E C} = Oc,,(y) for all m > n + 1. Since the family {Oc,f(y): e > 0 and C C A, C finite} is a base of the space X at y, we conclude that the sequence {Vn: n E N+} converges to y. Theorem 5.1 has been proved. Before deriving from theorem 5.1 some useful consequences, we give two easy results.

IV.5.2. Proposition.

Every closed subspace of a Preiss-.Simon compactum is itself a Preiss-Simon compactum.

IV.5.3. Proposition.

Every pseudocompact subspace of a I'reiss--Simon compactum X is itself a Preiss-Simon compactum, and is, moreover, closed in X.

Proof. Let Y C X, Y pseudocompact. By proposition 5.2 we may assume, without loss of generality, that Y = X. It now remains to prove that Y = X. Assume the contrary, and fix a point x E X \Y. There is a sequence {Vn: n E N+} of nonempty open sets Vn in X converging to x. Clearly, { Vn fl Y: n E N+} is then a discrete family of nonempty open sets in Y. But Y is pseudocompact, and hence such a family cannot exist in it. Thus Y = X.

IV.5.4. Theorem [137].

Every pseudocompact subspace of an arbitrary Eberlein compactum X is itself an Eberlein compactum, and is, moreover, closed in X.

Proof. This follows from theorem 5.1, proposition 5.3, and the fact that every compactum lying in an Eberlein compactum is itself an Eberlein compactum.

IV.5.5. Theorem [137]. Let X be a compactum, and Y a pseudocompact subspace of CC(X). Then Y is an Eberlein compactum, and is, moreover, closed in CC(X).

Proof. Denote by F the closure of Y in CC(X). By the theorem of M. O. Asanov and N. V. Velichko (see chapt. 111, §4), F is a compactum. Clearly, F is an Eberlein compactum. It remains to refer to theorem 5.4.

IV.5.6. Corollary [137].

Every pseudocompact subspace of a Banach space with the weak topology is an Eberlein compactum.

Proof.' It suffices to recall that every Banach space with the weak topology can be homeomorphically embedded in a space Cp(X) for some compactum X.

5. TI4C PRCISS-SIMON THEOREM

155

Yet another interesting result. related to Eberlein compacta can be obtained from the following general assertion about Preiss-Simon compacta.

IV.5.7. Proposition.

No Preiss-Simon compactum X can be the Stone-Cech

compactification of a subspace Y C X different from X.

Proof. Assume the contrary. Then there is a Y C X for which ,61' = X and Y:1- X.

Fix a point x E X \ Y, and pitt Z = X \ {x}. Then Y C Z C X, and 6Y = X implies /3Z = X. But ,3Z \ Z = X \ Z = {x}, i.e. the `remainder' of Z in 13Z consists of precisely one point. This implies that Z is pseudocompact. In fact, otherwise there would exist a continuous map f from Z onto some noncompact space M with a countable base, and for the extension f : /3Z OM of f to a continuous map between the Stone-Cech compact,ifications we would have 13M \ M C j ('6Z \ Z),

f (/)/, \ Z)j -- I while the set /3M \ Al is rclntrndieting die since the compactum [3M does not have a countable base. Thus Z is pseudocompact. But Z lies in the Preiss-Simon compactum X. Ilence, by proposition 5.3, Z is a compactum: X =/3Z = Z. We have obtained a contradiction with the fact that X \ Z # 0.

IV.5.8. Theorem. If X is an Eberlein compactum, and Y C X, Y

X, then X

is not the Stone-Cech compactification of Y. Proof. This follows from theorem 5.1 and proposition 5.7. E. A. R.eznichenko has constructed a Corson compactum X for which Cp(X) is K-analytic (and, hence, Lindel5f), and in which there is a point x such that. X is the Stone-Cech compactification of its subspace X \ {x}. By proposition 5.7, this compactum cannot be a Preiss--Simon compactum. The following characterization of Preiss-Simon compacta is of some interest.

IV.5.9. Theorem.

For a countably compact space X the following assertions are

equivalent:

a) X is a Preiss-Simon space; b) in X every pseudocompact subspace is closed; c) for any F C X which is closed in X and any point x E F which is not isolated in F, the space F \ {x} is not pseudocompact.

Proof. It has already been proved that a) implies b) (proposition 5.3). Clearly, c) follows from b). It remains to prove that c) implies a). So, let x E X, A C X, and x E A. Put F = A. If x is isolated in F, then x E A and x is isolated in A. Then the sequence {U,,: n E N+} where U. _ {x} for all n consists of open sets in A and converges to x. Let x be non isolated in F.

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Then, by c), the space Y = F\{x} is not pseudocompact. There is an infinite family = {V,,: n E N+} of nonempty open sets in Y which is discrete in Y. Consider an arbitrary neighborhood U of x in X. The set F \ U is closed in X, and thus countably compact. Hence there is an index m E N+ such that W,, = V !1 (F \ U) = 0 for all n > m, since otherwise the family 77 = {W,,: n E N+, 44;, 0} would be an infinite discrete family in F \ U, contradicting the countable compactness of F \ U. Thus the sequence {V,,: n E N+} converges to x. But then so does the sequence {V f1 A: n E N+}, and all V fl A are open in A and nonempty, since 4 = F D Y. The proof of the following theorem is based on theorem 5.9.

IV.5.10. Theorem.

Every condensation f : X --p Y from a pseudocompact space X onto some subspace of a countably compact Preiss-Simon space Y is a homeomorphism onto this subspace.

Proof. By proposition 5.3, f (X) is closed in Y. We may therefore assume that f (X) = Y. Let F be any closed set in X. There are open sets Ua, where a E A, such that F = fl{Ua: a E A}, since X is regular. Every Fa = Ua, being the closure of an open set in a pseudocompact space, is pseudocompact. Hence f (Fa) is a pseudocompact subspace of Y. Thus the set 7 = fl{ f (Fa): a E A} is also closed in

Y. But (D = n1 f (Fa): a E A} = f (fl{Fa: a E A}) = f (F), since f is bijective. So we have proved that the image under f of every closed set in X is closed in Y. Thus f is a homeomorphism.

IV.5.11. Corollary.

If a pseudocompact space X can be condensed onto some subspace of an Eberlein compactum, then this condensation is a homeomorphism, and X is an Eberlein compactum.

IV.5.12. Remark. Suppose that in X we can find for every point sequence {x,,: n E N} converging to a point x a sequence {U,,: n E N) of open sets converging to x and satisfying the condition: x,, E U for all n E N. Then X has a countable base at x. This becomes clear if we put x = x for all n E N. This observation sheds additional light on the definition of Preiss-Simon spaces.

6. Adequate families of sets: a method for constructing Corson compacta In this section we present a general construction, proposed by M. Talagrand, allowing us to obtain interesting examples of Corson compacta. Let T be an infinite set, and A a family of subsets of T satisfying the conditions: 1) UA = T, and if A E A, then all subsets of A belong to A; 2) if all finite subsets of a set A C T belong to A, then A E A. Such a family A is called adequate. Below, XA denotes the characteristic function of a set A in T: XA(x) = 1 if x e A

andXA(x)=0ifxET\A.

We associate two spaces with an arbitrary family A of subsets of a set T: the subspace KA = {x E {0,1 }T: {a E T: xa = 1} E A) of the product of T copies of

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the discrete colon D = {0,1}, formed by all characteristic functions XA of sets A E A,

and an object of `dual' type: the space T' = TT = T U {C}, where f T, the points of T are isolated in T', and the neighborhoods of C are the complements to unions of finite sets of elements of A. With these notations, which we adhere to in the whole section, the following general assertions hold.

IV.6.1. Proposition. If A is an adequate family of sets, then KA is compact. Proof. We show that KA is closed in the compactum VT. Let X' E DT \ KA. Then X' = XA' for some A' C T for which A' 0 A. Since A is adequate, there is a finite B C A' such that B 0 A. Then B is not contained in any element of A. This implies that the set W = {x E VT: x(B) _ {1}} does not intersect KA. Clearly, W is open and x' E W. Hence KA is closed in VT. A compactum K C DT is called adequate if K = KA for some adequate family A of subsets of T. By &A(6) we denote the function on KA that is identically zero. For t E T we define the function 6A(t) on KA with values in V = {0, 1} by bA(t)(x) = x(t) for all x E KA. Clearly, for all t E TA the functions 6A(t) are continuous. In this way we have defined a map bA: TA -+ CP(KA, D) C C,(KA)

We will use these notations throughout below, but for a fixed A we write 6, T', and K instead of bA, TA, and KA.

IV.6.2. Proposition. If A is an adequate family, then bA maps TA homeomorphically onto a closed subspace of the spaces CP(KA, D) and CP(KA) which separates the points of KA.

Proof. Let t E T. A base of the point 6(t) in b(T*) is formed by all sets of the form

W(bt,x1... xk) _ (f E b(T'):.f(xi) = St(xi), where i = 1,...,k}, x1 ...,xk E K. If t E T, then for x = X(t) we have St(x) = x(t) = 1, but b,(x)- 0 for all $ E T' \ {t}. This implies that all points of 5(T) are isolated in S(T'). If A,,-, Ak E A, then

W=T'\U{Ai: i=1,...,k}isaneighborhood off,and6(W)=W(b(C),xi.... xk), where xi = XAi for i = 1, ... , k. Thus we have proved that T' is homeomorphic to

b(T'). For X, y E K, x # y, we have X, y E VT, and hence there is a t E T for which x(t) # y(t), i.e. bt(x) # St(y). Therefore 6(T') separates the points of K. It remains to verify that b(T*) is closed in CP(K). Since CP(K,D) is closed in CP(K) and b(T') C CP(K,D), it suffices to prove that S(T') is closed in CP(K,D). We first establish that KI = or(PT) fl K is everywhere dense in K, where o(DT) = {x E DT:

Ix-'(1)) < Q. Indeed, let x E K. Then x = XA for some A E A. For any finite P C T we have XpnA E o.(VT) fl K and

p = x1 p. Hence 71 = K. The restriction map n: CP(K) -. CP(KI) is a condensation, since ICt = K. Thus it suffices to prove that rr(S(T')) is compact. Let W = W (f, xi, ... , xk) = {g E CP(K1i D): g(xi) = 0, i = 1, ... , k) be a standard neighborhood of the point f = 7rb(e)

IV. LINDELOP NUMBER TYPE PROPERTIES FOR. FUNCTION SPACES

158

in CP(K1iD). Each xi E K1 is a XB; for some finite Bi C T. Put B = U{Bi: i = 1,... , k}. Then for t E B we have zrbt(xi) = xi(t) = 0 for all i = 1,...,k, i.e. 7r(bt) E W. So, any neighborhood W off E ir(b(T')) contains all elements of ir(b(T`)) except finitely many. Thus ir(b(T')) is compact, and b(T*) = n"1 ir(b(T')) is closed in CC(K,V). Proposition 6.2 has been proved.

Proposition 6.2 allows us to identify in the sequel t and bt, and to assume that T* C CC(K, D) C C,(K). W e endow w = {0,1, 2, ... } with the discrete topology; the set of all continuous images of closed subspaces of the space (T' x w)" is denoted by PA. The class PA is countably multiplicative, a-additive, and closed under the operations of transition to a closed subspace or continuous image. Since the subspace TA C C,(KA, D) separates the points of KA, TA E PA, and the compactum KA is zero-dimensional, the results in §3 of chapt. 4 imply that

IV.6.3. Proposition.

If the family A is adequate, then Cr,(KA,D) E PA and

CP(KA) E PA, i.e. the spaces C,(KA, D) and CP(KA are continuous images of closed sets in the space (TA x w)". We also need the following characterization of zero-dimensional Eberlein compacta, which follows from the `intrinsic' characterization of Eberlein compacta obtained in §4.

IV.6.4. Theorem. Let X be a zero-dimensional compactum. Then X is an Eberlein compactum if and only if the space Cp(X, D) is a-compact.

Proof. By theorem IV.4.14, there is a family 13 = U{yi: i E N+} of nonempty open Fe-subsets of X, To-separating the points of X, and such that every yi is point finite. Every U E 13 is a-compact, and, moreover, a Lindelof space. Hence there exists a disjoint family µu of open-closed sets in X such that Upu = U. Putting ii = U{µu: U E yi} and B = {yi: i E N+}, we obtain a To-separating family 13 of open-closed sets in X, To-separating the points of X, where every family yi is point finite. For i E N+ we put Pi = {Xu: U E yi} U {X0} C CP(X,D). Since the family yi is point finite, Pi if- ^ ' ompact subspace of C,,(X, D). Hence the subspace Y = U{Pi: i E N+} is a-compact. Since 13 is a To-separating family, Y separates the points of X. But X is a zerodimensional compactum. Proposition IV.3.3 now implies that the space CP(X, D) is itself a-compact. Conversely, let CC(X, D) be u-compact. Since X is a zero-dimensional compactum, the subspace C,(X,D) of C,(X) separates the points of X. Hence X is an Eberlein compactum. Using theorem 6.4 we obtain

IV.6.5. Theorem [171.

Let A be an adequate family. Then I\..1 is an Eberlein

'rrimnmrtfsm if and rmly if TA is a e-rrmr`aet sr413CF.

6. ADEQUATE FAMILIES OF SETS

M

Proof. Let KA be an Eberlein compactum. The compactum KA lies in VT and is thus zero-dimensional. By theorem 6.4 the space Cp(KA, D) is a-compact. But by proposition 6.2 the space TA is homeomorphic to a closed subspace of Cp(KA, D). Thus TA is a-compact. Conversely, let TA be a-compact. By proposition 6.2 we may assume that TA C

Cp(KA, D) and that TA separates the points of KA. But then KA is an Eberlein compactum (see chapt. IV, §1). Here is yet another result of the same type.

IV.6.6. Theorem [17]. Let A be an adequate family. Then Cp(KA) is a Lindelof E-space if and only if TA is a Lindelof E-space.

Proof. If Cp(KA) is a Lindelof E-space, then so is TA, since TA is homeomorphic to a closed subspace of Cp(KA) by proposition 6.2. Conversely, let TA be a Lindelof E-space. We have TA C Cp(KA) in the sense of proposition 6.2, with TA separating the points of KA. Since KA is a compactum, theorem IV.2.10 implies that Cp(KA) is a Lindelof E-space.

Replacing in this reasoning the reference to theorem IV.2.10 by a reference to theorem IV.2.4, we obtain a proof of the following assertion.

IV.6.7. Theorem.

Let A be an adequate family. Then Cp(KA) is a K-analytic

space if and only if TA is a !C-analytic space.

We now consider two subtle examples of adequate families, as well as adequate compacta corresponding to them. The first example is due to M. Talagrand [149]. IV.6.8. Example. Consider the set E = w"' of all sequences of nonnegative integers, and the set S of all finite sequences of such numbers. For s E S we denote by Isl the

length of s, i.e. the number of terms in s. Ifs E S and a E E (and t E S), we put s -< a (respectively, s -< t) ifs consists of the first lit terms of a (of t). If a E E, then a n is the finite sequence formed by the first n terms of a.

Put T = E and B = {{a}: a E E}. Consider on E the usual Baire metric d: if a, p E E and a p, then d(p, a) = 1/n(p, a), where n(p, a) is the smallest n E w for which p I n 76 a I n. For n E N we denote by An the family of all subsets A C E in which for two arbitrary distinct elements the first n terms coincide, while the (n + 1)st

terms are distinct, i.e. A E A if and only if the distance between any two distinct points in A is 1/(n + 1). Clearly, the families B and An are adequate. We show that A = U{An: n E w} is also an adequate family. It is obvious that if A E A and B C A, then also B E A. Suppose that A C E and that every finite subset of A belongs to A. If A 0 B, then there are two distinct elements or and p in A. We denote by n' the smallest n E w such that p I n 0 a I n, i.e. n' = n(p,a). Then n' > 1 and p I k = a I k for k < n, while the definition of the family A,,. implies that if B E A and p,a E B, then B E A,,._I. In particular, every

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IV. LINDELOF NUMBER TYPE PROPERTIES FOR FUNCTION SPACES

finite subset of A containing p and or belongs to .A,,._1. By the definition of A,,._1 this implies that A E A,,._1. So A is an adequate family. Put L = Kg, K. = KA,,, and K = K,1. For m # n we clearly have An f A., = B. Hence K fl K,n = L. We show that the space T` = TA is not o-compact.

Let E be represented as E = U{T,,: n E w}. We endow E with the topology generated by the Baire metric d. The metric space (E, d) is complete [66), hence the topological space E has the Baire property. Consequently, there is an n' E w for which the interior of the closure of the set Tn, is nonempty. This means that {Q: s' -< v} C Tn,, for some finite sequence s'. For each q E w we denote by s' q the sequence of length ,s'I + 1 which is obtained if q is adjoined to the sequence s' as last term. The set Vq = for E E: S' q -< o} is nonempty, open, and is contained in the closure of the set T,,.. Hence T,,, fl V9 # 0, whatever q E W. For each q E w we fix a Oq E T,,. such that

oq. Put A = {oq: q E w}. Clearly, Then A is closed in TA. Since A C T, the subspace A is A E A,.,, C A and A C discrete. Using the fact that A is infinite, we conclude that the subspace Tn, U {C} of TA is not compact. Consequently, TA is not a o-compact space. By theorem 6.5 this implies that KA is not an Eberlein compactum. Note that, since all points of TA except one are isolated and TA is not or-compact, TA does not contain an everywhere dense or-compact subspace, i.e. TA is not k-separable.

We show that every compactum K is an Eberlein compactum. By theorem 6.5 it suffices to establish that the space T, = TA. is or-compact.

Take an arbitrary finite sequence s E S of length n + 1, and put F, = {Q E E: s -< a} U {C}. Each set A E An intersects F, at at most one element, since the (n + 1)st terms of two arbitrary distinct elements of A are always distinct. Therefore, for an arbitrary neighborhood U of i in TA, the set F. \ U is finite. Consequently, the subspace F, of TA., is compact, for every finite sequence s E S of length n + 1. Clearly, TAn = U{F,: s E S and Isl = n + 1}. Since the set of all finite sequences s E S of length n + 1 is countable, we conclude that the space TA. is or-compact. We have K=U{K,,: nEw}, since A=U{An: n E w}. Thus, K is the union of a countable set of Eberlein compacta. Every set A E A is countable. In fact, A E An for some n E w, and hence the first n terms of all or E A are identical. If A would be uncountable, it would contain two distinct elements with identical (n + 1)st terms. But this contradicts the fact that A E An. The countability of all A E A implies that K lies in the E-product of DTI copies of the colon D = 10, 1}. Thus K is a Corson compactum. So, the example of the compactum K = KA convinces us that not every Corson compactum is an Eberlein compactum, and that the class of Eberlein compacta is not a-additive in the class of compacta.

Based on the properties of the objects constructed in example 6.8, we now prove an important assertion, due to R. Pol [133]

IV.6.9. Theorem.

Let P be a class of topological spaces having the following two

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161

properties:

a) if X E P, then every closed subspace of X belongs to P; b) if X E P, then every continuous image of X belongs to P. Then there is either an Eberlein compactum X such that CC(X) f P, or a compactum Y which is not an Eberlein compactum such that Cp(Y) E P. Proof. We need the following

IV.6.10. Lemma.

Let {X;: i E N+} be a countable family of Eberlein compacta. Then the product fI{CC(X;, I): i E N+}, when I = [0,1] is an interval, is homeo-

morphic to a closed subspace of the space Cp(Y), where Y is an Eberlein compactum.

Proof of the lemma. We assume that the compacta X,, i E N+, are pairwise disjoint. Take their free topological sum X = Ee{X;: i E N+}, and compactify the locally compact space X by a new point b. We obtain an Eberlein compactum Y = X U {b} (the conditions of Rosenthal's criterion (see chapt. IV, §1) are trivially fulfilled). Clearly, the space fI{CC(Xi, I): i E N+} is homeomorphic to the closed subspace

if E Cp(Y): f (b) = 0 and 0 < P x) < 1/i for all x E X i E N+} of Cp(Y). Lemma 6.10 has been proved. The spaces TA and TAr consist of the same point set. We denote by Tn the topology of Ti,,, and by T the topology of TA. Clearly, T is the lattice union of the topologies Tn with respect ton E w, i.e. T is the smallest topology on TA containing all topologies T.

This means that the diagonal AT; = {(t,t,... ): t E T'}, considered as a subspace of the product H{TA,: n E w}, is homeomorphic to TA (the homeomorphism is realized

by projecting (t, t, ...) - t). We show that the diagonal AT' is closed in the space Z = fI{TA, : n E w}. Take an

arbitrary point z = (t1, t2.... ) E Z not belonging to AT . Then tk tm for certain Then k, m E w, k # m. Either tk or tm is distinct from . Suppose, e.g., tk Vk = {tk} is an open set in TA,, and Vm = T* \ {tk} is an open set in Ti,,,; moreover, tk E Vk and tm E V,,. The set W of all points (t;, t2, ...) E Z such that tk E Vk and t;,, E Vm is open in Z, contains z, and does not intersect the diagonal DTA. In this way we have proved the following assertion.

IV.6.11. Proposition. The space TA is homeomorphic to a closed subspace of the product fI{TA,: n E w}. Since TA contains an infinite closed discrete subspace, the space (TA x w)' is homeomorphic to a closed subspace of (TA)W. Propositions 6.3 and 6.11 imply that the space CC(KA) is a continuous image of a closed subspace M of the space (ff{TA : n E w})W. But TAn is homeomorphic to a closed subspace of CC(Kn, I). Since all Kn are Eber-

lein compacta (see example 6.8), lemma 6.10 implies that the space (ff{Cp(Kn, I): n E w})W is homeomorphic to a closed subspace of the space CC(Y), with Y an Eberlein compactum. Hence M is also homeomorphic to a closed subspace of CC(Y). We now assume that the following assertion holds: A compactum X is an Eberlein

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IV. LINDEI.OF NUMBER TYPE PROPERTIES FOR FUNCTION SPACES

compactum if and only if C,,(X) E P. Then C,,(Y) E P, and a) implies that M E P, while h) implies that Cp(KA) E P, i.e. we find that KA is also an Eberlein compactum, contradicting our assertions in example 6.8. Theorem 6.9 has been proved. In the course of proving theorem 6.9 we have established that the space Cp(KA) is a continuous image of a closed subspace of the space (jj{TA.: n E w})-. We have seen that all spaces TA,, are or-compact, n E w. But the product of a countable set of spaces, as well as any closed subspace of such a product, is K-analytic 131, and, hence, a Lindelof E-space. So we have proved

IV.6.12. Proposition.

The space Cp(KA), where KA is the compactum of example 6.8, is 1C-analytic, and, hence, a Lindelof E-space. Thus, theorem IV.2.5 concerning the K-analyticity of the space Cp(X) for any Eberlein compactum X has no converse. Note that this conclusion can also be obtained as a formal consequence of theorem 6.9, since the class P of all IC-analytic spaces has the properties a) and b) in theorem 6.9. The following example is due to G. A. Sokolov.

IV.6.13. Example. Let T be a set of real numbers of cardinality RI, well ordered by some relation -<. We denote by < the usual order relation on T by magnitude. By A, we denote the family of all subsets of T on which the orders -< and < coincide, and by A2 we denote the family of all subsets of T on which the orders -< and < are opposite (i.e. -< is the inverse (opposite) of <). As can be readily verified, the family A = Al U A2 is an adequate family of subsets of T. We now need the following simple

IV.6.14. Lemma.

Every subset A of the set R of real numbers which is well ordered by magnitude using the relation < is countable.

Proof of the lemma. For each a E A which is not the largest element of A we define the element a+ E A as the smallest of the elements of A that are larger than a. We obtain a disjoint family y = {(a,a+): a E A} of nonempty intervals on the line R. The family -y is countable, hence A is countable. Note that lemma 6.14 remains valid if < is replaced by the relation opposite to it. Lemma 6.14 and this remark imply that all elements of A are countable sets. This

implies that the compactum KA lies in the E-product of ITI copies of the colon D = {0, 1}, i.e. KA is a Corson compactum. We now show that every infinite set A C T contains an infinite subset B belonging to A. Since A is infinite, it cannot be well ordered by either the relation < or the opposite relation <-1. Hence (A, <) contains either an infinite increasing sequence or an infinite decreasing sequence. We consider the second case in more detail (in the first case the reasoning is completely similar). So, let rl: a1 > a2 > ... be an infinite decreasing sequence of elements of A.

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Denote by ak, the smallest term of this sequence in the sense of the well order -<, i.e. ak, = mint {al, a2.... }. By ak, we denote the smallest term of the set {a,,: n > k1 + 1 } in the sense of -<. Repeating this construction in the obvious way by induction, we obtain a sequence ak,, ak2, ... for which k1 < k2 < - ; this implies that ak, > ak, > -

. We have obtained an infinite set .... Moreover, by construction, ak, -< ak, -< B = {ak,, ak..... } C A on which -< coincides with <-'. Hence B E A2 C A. In the first case we similarly construct a set B = {ak ak ... } C A on which -

coincides with <. Then B E .A:1 C A. Since every element of A is, by the definition of the topology in TA, a closed discrete subspace of TA, the proof given above implies that every infinite set A C TA contains an infinite closed discrete subspace of TA. In particular, all compacta in TA are finite. From this we infer that TA is not a Lindelof E-space, which implies, by theorem 6.6, that CpKA) is also not a Lindelof E-space. Since TA is an uncountable set, it remains to prove the following assertion.

IV.6.15. Proposition. If Y is a Lindelof E-space in which all compacta arc finite, then Y is countable.

Proof of proposition 6.15. Each Lindelof E-space is a continuous image of a Lindelof p-space (see chapt. 0). Every p-space is a k-space, and every Lindelof space X has the property: every disjoint open cover of X is countable. Spaces having this property (we have come across them already in chapt. 11) are called o-Lindelof spaces. Besides the Lindelof spaces, all spaces with a countable Suslin number, all connected spaces, and all weakly Lindelof spaces (i.e. spaces in which any open cover contains a countable subfamily covering an everywhere dense set in the space) are o-Lindelof spaces. Therefore proposition 6.15 is a particular case of the following proposition.

IV.6.16. Proposition.

Let Y be an uncountable space in which all compacta are finite. Then Y cannot be represented as a continuous image of an o-Lindeldf kit-Space (see chapt. 0). Proof. Assume the contrary. Let Y be the image of an o-Lindelof kR,-space X tinder

a continuous map f. Take an arbitrary point y E Y. We show that the closed set f-'(y) is open in X. Assume the contrary. Then the characteristic function gA of the set A = X \ f-'(y) is not continuous, and since X is a km-space, the restriction of 9A to some compactum C X is a discontinuous function on 4P. The set gA' (0) fl = f -' (y) fl is closed. Hence the set gA' (1) fl 4, = 4 \ f -I (y) is not

closed, i.e. there is a point xl E f-'(y) such that xl E F \ f-'(y). But then the point y = f (xl) lies in the closure of the set f (4') \ {y}, and hence the compactum f (f) is infinite, contradicting the conditions. So all sets f `(y) are open in X; together they form a disjoint open cover of X. Since

X is an o-Lindelof space, the family {f-'(y): y E Y} is countable. But f-'(y) 34 0 for all y E Y. Consequently the set Y is countable. We have thus established that KA is a Corson compactum for which Cp(KA) is not

IV. LINDELOF NUMBER TYPE PROPERTIES FOR FUNCTION SPACES

164

a Lindelof E-space. Since a closed subspace of a Lindelof k-space is a Lindelof k-space, proposition 6.16 and the reasoning given above imply that Cp(KA) is not a continuous image of any Lindelof k-space, although it is a Lindelof space. We have thus proved

IV.6.17. Proposition. There is a Corson compactum X for which Cp(X) is not a continuous image of any Lindelof k-space.

7. The Lindelof number of the space Cp(X), and scattered compacta In this section we consider an interesting example of a scattered compactum X for which Cp(X) is a Lindelof space, and give a remarkable theorem of R. Pol which

implies that a separable scattered compactum with the similar property does not exist.

In particular, we show that not every compactum X for which Cp(X) is Lindelof need be a Corson compactum.

IV.7.1. Example (see [156], R. Pol [132]).

Let 0 be the set of all countable ordinals, I' the set of all nonlimit ordinals in 11, and S = S2 \ r the set of all limit ordinals in 52. For each A E S we fix a sequence sa: w --' I such that s,\(n) < A for all n E w, while s,\ converges to A with respect to the order topology on 11. In 0 we introduce a new topology, as follows: the points in r are taken to be isolated, while sets of the form {A} U {sa(n): n > m} are taken to be basic neighborhoods of a point A E S. We obtain a locally compact space S2, and denote its one-point compactification by X: X = S2 U {w1}. There is a countable family 77 of subsets of S such that if A1, A2 E S and Al 0 A2,

then there is a P E 77 for which Al E P and A2 0 P. To obtain such a family 11, it suffices to bijectively map S onto a subset of the real line R, and to take the preimages

under this map of some countable basis of R. Put £ = {{x}: x E r}U{Pur: P E 771. Clearly, E is a o-disjoint family of open sets in X which To-separates the points of X. At the same time, from Fodor's lemma [77] we can in a standard manner derive that X does not contain a point-finite family of open sets of type Fo which To-separates the points of X. Hence X is not a Corson compactum. Clearly, X is a monolithic and scattered compactum. Moreover, it is zero-dimensional.

We show that Cp(X) is a Lindelof space. By proposition IV.3.7 it suffices to establish that the space (Cp(X, D))"0 is Lindelof.

Put 1) E = If E C(X, D): If-' (1) n SI < 1}; and 2) G = If E C(X, D): f (wl) = 0}. We prove the following assertion:

a) for each f E G there are fl,. .. , fm E E such that f = fi + + fM. Let f E G and f -'(1) n S = {A,, ... , Ak}, with A, 4 Al if i # j. For i < k we take f; E E such that fi(A;) = 1. Then fk+1 = f = fi++fk.

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So a) has been proved. We now prove that E has the strong condensation property (see chapt. 4, §2).

Put Eo = { f E C(X, D): f '(1) C r} C E. We first show that E0 is a space with the strong condensation property. Let A C Eo and ]AI =111. For each f E A the set Sj = f-'(1) is finite, hence there are an uncountable set C C A and a finite set K C I' such that S1 n s, = K for any

two distinct f,g E C. Take the characteristic function h of the set K. Clearly, for any neighborhood V of h the set C \ V is finite. So, Eo has the strong condensation property. Consider now an arbitrary set A C E of cardinality 121i and put E = {A E S: there

is an f E A for which f()) = 1}. Two cases are possible. Case 1. The set E is bounded in the ordered set Sl, i.e. E C 10, a] for some a < w1. If IA fl Eo) = 141, then we can use the strong condensation property of Eo. Suppose A fl Eo is countable. Then there are a a E [0, a] and an uncountable set B C A such that f (A) = I for all f E B. Let g E E and g(.\) = 1. Then If - g: f E B} C Eo,

and, as has been proved above, there are an uncountable set C1 C If - g: f E B} and a point xi E Eo such that the set C1 \ V is finite for any neighborhood V of x1. Clearly, any neighborhood of x = xi + g E E contains all points of the set C = {q+ g: 0 E Ci} C A except finitely many. Case 2. The set E is not bounded in the ordered set Q. By transfinite recursion we can then construct, in an obvious manner, a transfinite sequence {A0: a < wi} C E such that for each a < wi the inequality

Aa=sup{\p: ,Q
Fix for each a < w1 a function fa E A such that fa 54 fp if a # /3). Put J. = (µa, A, ] for a < wi; clearly, Ja fl Jp = 0 for a # P. Consider the sets Ha = fat (1) fl Ja and Fa == f;' (1) \ Ja. The sets F. are finite, and Ha fl Hp = 0 for any two distinct a and /3. Denote by ha and 9a the characteristic functions of the sets H. and Fa, respectively. Then ha E E, ga E Eo, and fa = 9a+ha. Clearly, every neighborhood of the function which vanishes identically contains all elements of the set {ha: a < wi }, except finitely many.

Since {ga: a < w1} C Eo, by the above there are an uncountable set of ordinals 0 C St and a function 0 E E0 such that for every neighborhood V of ¢ the set {a E 0: ga 0 V} is finite. This implies that every neighborhood of 0 E E contains all elements

of the uncountable set C = If,,: a E 91, except finitely many. Thus E is a space with the strong condensation property. This implies that E'' is a Lindelof space (see proposition IV.2.21). We now need

IV.7.2. Lemma.

Let G be an Abelian topological group, and E a subspace of it

such that E' is a Lindelof space. Also, let for each a E G there be a1,... , an E E such that a = ai + + am. Then G' is a Lindelof space.

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Proof of the lemma. The space E' x w' is Lindelof, since either E is compact or contains an infinite closed discrete subspace. The map f : EW x w"' -+ G" is defined by f ((al, a2, ... ), (MI, M2.... )) = (a1 +, *, + ani)am,+1 + ... }amt+m2, ... }. Clearly,

f is continuous,and f (E" x w") = G. We continue with the consideration of example 7.1. Lemma 7.2 can be applied

to the case when E = {f E C(X,D): If-'(1) fl Sj < 1} and G = {f E C(X,D): f (wi) = 0} (see 1) and 2)). In fact, by the above, E' is a Lindelof space; G is clearly an Abelian topological group. Moreover, the last assumption in lemma 7.2 is also fulfilled. Applying this lemma, we conclude that GW is a Lindelof space. We now note that the space C,,(X, D) is (canonically) homeomorphic to the space G x D. Hence (CC(X, D))" is homeomorphic to the space G" x D", and it is a Lindelof space since D" is compact. So, we have proved that X is a zero-dimensional (even scattered) compactum which is not a Corson compactum, although it is monolithic and has a o-disjoint To-separating family of open sets, for which Cl,(X) is a Lindelof space.

It is appropriate to compare the last example with the following result, which is a consequence of a theorem of R. Pol and D. P. Baturov, and which gives a large amount of zero-dimensional compacta X for which Cp(X) is not a Lindelof space and, consequently (chapt. I, §5), not normal.

IV.7.3. Theorem. Let X be an uncountable separable scattered compactum whose with derived set is empty. Then CD(X) is not normal. Proof. By D. P. Baturov's theorem (chapt. III, §6) it suffices to prove that C,,(X) is not Lindelof. But Cp(X) contains a closed subspace homeomorphic to CP(X,D"), and the latter is homeomorphic to Cp(X,V) x (CP(X, D"))", and hence contains a closed subspace homeomorphic to CP(X, D) x w". Therefore it suffices to verify that CD(X, D) x w" contains an uncountable closed discrete subspace. As is well known, the derived set XtO'> of a set X is defined by transfinite recursion,

as follows: Xt°> = X; if a is a limit ordinal, then X(a) = n{Xt#>: 0 < a}; X("+'> is the set of all nonisolated points of X(°>. All Xt°`> are closed in X. Hence, if X is a compactum and V"1) = 0, then Xt"O> = 0 for some a° < wi. It is obvious that in this case X = U{ X(°`> \ X("+'): a < a°}. Since X is uncountable, X(Q) \ X(c'+') is uncountable for some a < a°. We denote by C the first a < a° for which X (a) \ X(a+'> is uncountable. The separability of X implies that C > 1. Put F = Xt{>. The set M = X \F is clearly countable, while the set A = F\F('> of all nonisolated points of F is uncountable. Moreover, F C M, since X is a scattered compactum (already Xt°> \ Xt'> is everywhere dense in X). It is now sufficient to verify the following

IV.7.4. Proposition. Let X be a zero-dimensional compactum, M C X, M open, countable, and everywhere dense in X. Moreover, let the set A of all isolated points of the space F = X \ M be uncountable and everywhere dense in F. Then the space

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C,,(X,D) x w' contains an uncountable closed discrete subspace.

Proof of proposition 7.4. For each a E A we fix an open-closed set Va such that V,, fl F = {a}, and denote by fa the characteristic function of Va. Consider the sets G = If.: a E Al, L = If E C(X, D): f IF, = 0}, and S = LUG. The set L is closed in CC(X,D), and contains all points of CC(X,D) that are limit points of G. In fact, if f E C,,(X,D) and f E G, then for each a E A there is a b E A such that b 0 a and fb(a) = f (a). But a b and a E A imply that a Vb, hence fb(a) = 0. Thus f IA - 0, and then also J IF = 0, since A is everywhere dense in F. So f E L. We conclude that S is closed in CP(X, D), and that G is an open and discrete subspace of S. Every function f E S can be put in correspondence with its restriction onto Al; the corresponding map is denoted by u: S --+ DM. We denote by To the set of all functions 1: M -+ D, not necessarily continuous, for which there exists a neighborhood V, of F in X such that t vanishes on precisely the set 1, fl M. We put 7', = DM \To.

If f E L, then F C f '(0), and V = f '(0) is a neighborhood of the set F in X on which f vanishes identically. Hence u(L) C To. On the other hand, if f E G, then f is not equal to zero at some point x in F, and since F C M, every neighborhood of x contains a point y in M for which f (y) = 1. This implies that u(G) C T1, whence DM: G = u -(TI). But u is G is the complete preimage under the map u: S continuous. Hence the graph Gr(u) = {(f, u(f )): f E G} of the map ula is a closed set in the space S x T, [66). Since G is discrete and uncountable, the subspace Gr(u) of S x T, is also discrete and uncountable. The set M = X \ F is countable. Hence F is a set of type G6 in X. Since F is closed in X and X is a compactum, there is a countable family {V,,: n E w} of neighborhoods of F in X which is a base of F in X, i.e. is such that every neighborhood of F contains some V,,.

Put P = It E DM: tlv nM = 0} for n E W. Clearly, Pn is closed in DM, and To = U{Pn: n E w}. Consequently, To is a set of type F,, in DM, and T, = DM \To is a set of type G6 in DM, i.e. Ti = 1 {Un: n E w}, where each Un is open in DM. But then (see chapt. 0) the space T1 is homeomorphic to a closed subspace of the product 11{U,,: n E w}. Each U,,, being a locally compact zero-dimensional space with a countable base (M is countable), is homeomorphic to a closed subspace of w"' 166). Hence T1 is homeomorphic to a closed subspace of (w')"= w" So, the space CC(X, D) x w"' contains a closed subspace homeomorphic to S x T, (recall that S is closed in CC(X, D)), while the latter, as has been shown above, contains an uncountable closed discrete subspace. Proposition 7.4 and, with it, theorem 7.3, have been proved. The compactum X considered in example 7.1 is scattered, and already its third derived set X(9) is empty. In example 7.1 we have proved that the space C,(X) is Lindelof. This does not contradict with theorem 7.3: X is not separable. Thus, the assumption in theorem 7.3 that the compactum is separable is utmost essential. Formally this assumption can be somewhat weakened, replacing it by the requirement

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that the Suslin number is countable: the Suslin number of a scattered compactum is countable if and only if the compactum is separable.

Under the continuum hypothesis, K. Kunen has constructed a scattered compactum K of cardinality 81 every finite power of which is hereditarily separable. By theorem 11.5.10, every finite power of the space CC(K) is hereditarily Lindelof, i.e. the assumption in theorem 7.3 that the w1th derived set is empty is essential. E. A. Reznichenko has found an important application of proposition 7.3. A somewhat simpler version of his result is given in the following section.

8. The Lindelof number of CC(X) and Martin's axiom In this section we need certain results based on the constructions in chapt. IV, §3. We begin with a general auxiliary assertion.

IV.8.1. Proposition.

Let X be a compactum. Then the space Cp(X") is a continuous image of a closed subspace of the product of the space (CI,(X))'" and a compactum.

Proof. Consider the projection at: X" - X, where ri(y) = y(i) E X for all y E X", and put F, = 7rq(Cp(X)) C Cp(X"), i E w. Thus, Pi = If o ir,: f E Cp(X)} is a subspace of Cp(X") homeomorphic to Cp(X). Clearly, the subspace S = U{Fi: i E w} of Cp(X") separates the points of X", and is a continuous image of Cp(X) x w. The space Cp(X) contains an infinite closed discrete subspace. Hence the class Px of all spaces representable as a continuous image of a closed subspace of the product of the space (CC(X))' and a compactum, is no-perfect (see §2), and S E Px. Since X" is compact, theorem IV.2.9 implies that Cp(X") E Px, as required. In the case of zero-dimensional compacta, proposition 8.1 can be substantially improved upon. Namely, invoking proposition 3.3, the proof of proposition 8.1 can be easily transformed into a proof of the following proposition.

IV.8.2. Proposition.

If X is a zero-dimensional compactum, then the space Cp(X/") is a continuous image of the free sum Ee{(Cp(X))": n E w} of the spaces (Cp(X ))n.

Of course, the conclusions of assertions 8.1 and 8.2 remain true if X" is replaced by X", where n E w. Proposition 8.1 has the following consequences.

IV.8.3. Theorem. Let X be a compactum, and (CC(X))" a Lindelof space. Then Cp(X") is a Lindelof space (as are also all spaces Cp(X") for all n E N).

IV.8.4. Theorem.

Let X be a compactum, and Cp(X) E P, where P is an

no-perfect class of spaces. Then also Cp(X") E P.

8. THE LINDEU%F NUMBER OF Cp(X) AND MARTIN'S AXIOM

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A strengthening of theorem 8.3 for the case of zero-dimensional compacta is the following

IV.8.5. Theorem.

Let X be a zero-dimensional compactum. Then: a) there is

a closed subspace P1 C CC(X,V"0) C CC(X) which can be continuously mapped onto Cp(X"'); and b) there is a closed subspace P2 C Cp(X,D"O) C CC(X) which can be continuously mapped onto (Cp(X))".

Proof. We may regard D"° as the Cantor perfect set in the interval I = [0, 1]. Then Cp(X, V10) becomes a closed subspace of the space Cp(X, I), which, in turn, is closed

in CC(X). By lemma 3.7, CC(X,D"°) can be continuously mapped onto C9(X,I). Since X is infinite, C9(X) is not r-countably compact (see chapt. I, §2). But CC(X) is the union of countable many subspaces homeomorphic to C,(X, I). This implies that the spaces C,,(X, I) and C,(X, D"°) are not countably compact. Thus, there is all infinite closed discrete subspace in C,(X,V"°). Clearly, CC(X,D"0) is homeomorphic to C,(X, D"() x C,(X, WO), since D"0 x D"0 = D"O. Therefore Cp(X, D"O) contains a closed subspace P1 homeomorphic to Cp(X,V"0) x w x w. Applying lemma 3.7 again and using the fact that for every n E w the space (Cp(X,D"0))" is homeomorphic to Cp(X, D"0), we arrive at the following conclusion: for any n E w the space CC(X, D110) can be continuously mapped onto the space (CC(X, I))". But (Cp(X))" = Cp(X, R") is the union of countably many subspaces homeomorphic to (CC(X, I))" = C,(X, I"), since X is a compactum. Therefore Cp(X,D"D) x w can be continuously mapped onto (Cp(X))", and the space P1 = Cp(X,D"0) x w x w can be continuously mapped onto e{(Cp(X))": n E w}. The latter space can be continuously mapped onto the space Cp((w), by proposition 8.2. Assertion a) has been proved. We prove b). Since X is a compactum, the space Cp(X) is a continuous image of Cp(X, I) x w. Using the fact that X is zero-dimensional and applying lemma 3.7, we

conclude that Cp(X, D"0) x w can be continuously mapped onto CC(X). But then CC(X,V"0) x co' can be continuously mapped onto (Cp(X))'". Since Cp(X,D"0) is homeomorphic to its countable power and contains an infinite closed discrete subspace, there is a closed subspace P2 in Cp(X,D"O) C C,(X) homeomorphic to CC(X, D"°) x w'". The subspace P2 in (CC(X))" is the one looked for. The following assertion was noted by A. V. Arkhangel'skil and E. A. Reznichenko.

IV.8.6. Theorem.

Let X be a zero-dimensional compactum. Then the following assertions are equivalent: a) the space C,,(X, D"O) is normal; b) the space C,(X, I) is normal; c) the space CC(X) is normal; d) the space CC(X) is Lindelof; e) the space (CC(X))" is Lindelof; f) the space CC(X"') is Lindelof.

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170

Proof. Clearly,

Since X" contains a compactum homeomorphic

to X, f) implies d). It remains to prove that a)=f). Then also d)=f), and all conditions a)-f) are equivalent. Let CP(X, DN0) be normal. By theorem 8.5, a certain closed subspace PI of CP(X,DN0) can be mapped onto CP(X"). The space CC(X,DNO) x CP(X,DNO) is also normal. By Corson's theorem, CP(X, DNO) is collectionwise normal. By Baturov's theorem, C,(X,DNO) is Lindelof. Hence the spaces P1 and CP(X") are Lindelof. Theorem 8.6 has been proved. We now prove the following theorem of E. A. Reznichenko, given in a somewhat weaker formulation.

IV.8.7. Theorem. Assume MA+-'CH, Martin's axiom combined with the negation of the continuum hypothesis. Then every compactum X for which the space (CP(X))" is normal, is N0-monolithic.

We immediately note that by Baturov's theorem (see chapt. ili, §6) we may assume that (CP(X))'' is a Lindelof space. The proof of theorem 8.7 is based on several lemmas. The first one is well known.

IV.8.8. Lemma. Assume MA + -NCH. Let L be a countable set, and let A, 8 be families of subsets of L satisfying: a) IA U BI < 2N0;

b) IA fl B< N0 for allAEA, BE B; c) all elements of A are infinite sets. Then there is a set M C L such that jAnMj = loo for all A E A, and IBnMI < No

for allBES. Using this lemma we can prove the following lemma, which is due to E. A. Reznichenko.

IV.8.9. Lemma. Assume MA + -CH, and let X be a compactum of weight 81 in which are given a countable everywhere dense set L and a nowhere dense closed set F. Then there is a set M C L such that M \ M = F and all points of M are isolated in the space M = F U M.

Proof. The set L1 = L \ F is everywhere dense in X, since F is nowhere dense in X. Now w(X) < N1 implies that F contains an everywhere dense subset P of cardinality < N. It also implies that there is a family y of open sets in X which contain F such that jryj < N1 and such that each neighborhood of F in X belongs to some element of '1

Since L1 is countable and everywhere dense in X, while X(X) < w(X) < ft1, for each point x E P there is a sequence C. = (Y,,: n e N} of points of LI converging to x. We fix for each x E P such a sequence tz, and denote by Ay the set of points of the sequence G. Since LI n F1 = 0, all sets A. are infinite. Put A = {A2: x E P}, and

8. THE

NUMBCR OF Cp(X) AND MARTIN'S AXIOM

13 = {LI \ V: V E y}. Then JAI S IPI < 81, IHI <_ IyH <_ 8I, and, since ,

171

converges

to x E P C F, the set A2 \ V is finite for every V E y. Hence IA fl PI < 8o for all sets A E A and B E B. Thus, all assumptions of lemma 8.8 are fulfilled. Applying this lemma, we conclude that there is a set M C Li such that for all x E P the set M fl A. is infinite, while the set M \ V is finite for all neighborhoods V of F in X. But now it is obvious that F D 11.1 \ M D P. Since P = F, we thus obtain

FDM\MDF,i.e. TIT \M=F.

Note that all points of it'l are isolated in M U F = M, since M fl F = 0 and only finitely many points of M can lie outside any neighborhood of F. Lemma 8.9 has been proved. We now need the following version of propsition 7.4, differing from this proposition most of all by the absence of the assumption about zero-dimensionality.

IV.8.1O. Proposition. Let X be a compactum, M a set of isolated points which is countable and everywhere dense in X. Suppose also that the set A of isolatcd points of the space l'' - X \ A4 is uncountable and everywhere dense in P. Then the space CF(X,D) x wW contains an uncountable closed discrete subspace.

Proof. Consider the compacturn Y obtained from X by glueing at one point y' all points of the set 4P = F \ A. Denote by w: X Y the canonical quotient map. Then = rr-'ir(
tains an uncountable closed discrete subspace. But ir: X - Y is a quotient map, hence Cr,(Y, D) is homeomorphic to a closed subspace of Cc(X, D). This implies that Cp(X, D) x w" also contains an uncountable closed discrete subspace.

IV.8.11. Proposition.

For each nonmetrizable compactum there is a continuous map onto a nonmetrizable compactum of weight RI.

Proof. Let X be a nonmetrizable compacturn. Then Cr,(X) is not separable, hence contains a left subspace F of cardinality NI. The diagonal product 0 = AF of the functions in F is a continuous map from X onto a compacturn Y in R"'. The dual map 0' going in the opposite direction maps CC(Y) homeomorphically onto a subspace

of Cc(X) containing the uncountable left subspace F (see chapt. 0). Hence C',(Y) is not hereditarily separable, and, moreover, does not have a countable network. We obtain w(Y) = nw(Y) = nw(Cz,(Y)) > 81 (see chapt. I). But Y C R"' implies that. w(Y) < xi. So w(Y) = l'lI. Now we have at our disposal everything we need to prove theorem 8.7. Proof of theorem 8.7. Let X be a compactum for which (C'(X))0 is a Lindelof space.

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Consider an arbitrary countable set T C X and its closure Y = T in X. The space (CC(Y))'" is Lindelof, since CC(X) can be continuously mapped onto CC(Y). Hence it suffices to prove that if the compactum X satisfies the assumptions of theorem 8.7 and is also separable, then it is metrizable. Assume the contrary. By proposition 8.11, X can then be continuously mapped onto a nonmetrizable compactum Y of weight 121. The compactum Y is also separable, and satisfies the assumptions of theorem 8.7, since under the dual map Cp(Y) is mapped onto a closed subspace of CC(X).

We put Z = Y x Y, and prove that the compactum Z is perfectly normal; by a well-known theorem [16], [661, this will imply that Y is metrizable, giving the required contradiction. Here also we assume the contrary: suppose Z is not perfectly normal. By a funda-

mental theorem of Z. Szentmiklossy, under the assumption MA+-'CH the spread of Z is uncountable. Thus Z contains an uncountable discrete subspace A. Put F = 7, and denote by U the interior of the compactum F in Z. All points of A fl u are isolated in F, and, moreover, isolated in U. But then all points of the set Aft U are also isolated in Z. The compactum Z is separable, since Y is. This implies that A n U is countable. Put Al = A \ U, and F1 = XI. It is now clear that the compactum F1 is nowhere

dense in Z, and that Al is an uncountable set of isolated points in F1 which is everywhere dense in Fl. All assumptions of lemma 8.9 are fulfilled for the compactum Z and set F1 C Z.

Applying this lemma, we conclude that there is a countable set M C Z for which M \ M = F1, while all points in M are isolated in the space M. Put ZI = M. Since (CC(Y))" is Lindelof, theorem 8.3 implies that the space Cp(Z) = Cp(Y2) is Lindelof. But then the space Cp(Z1) is Lindelof. On the other hand, proposition 8.10 may be applied to the compactum Z1 (in the role of X) and set F1 (in the role of F). This proposition implies that the space Cp(Z1i D) x w" contains an uncountable closed discrete subspace, and, hence, that Cp(Z1iD) x J`' is not Lindelof. From this we infer that Cp(Z1,DN0) is not Lindelof. This would contradict the fact that Cp(Z1) is Lindelof, since Cp(ZI, V'0) is homeomorphic to a closed subspace of Cp(Z1). So, it remains to prove the following assertion.

IV.8.12. Lemma. If a space X is such that the space Cp(X, D) x w`" is not Lindelof, then the space Cp(X,D"0) is not Lindelof. Proof. Assume the contrary, suppose CP(X, DK0) is a Lindelof space. Then Cp(X, DN0) is not countably compact, for otherwise Cp(X, Dx0) would be compact, so that Cp(X, D)

would be compact, and the product Cp(X, D) x wW would be a Lindelof space, contradicting the requirements. Hence Cp(X,DsO) contains an infinite closed discrete subspace. But CC(X, D"0) is homeomorphic to its own countable power, since Dx0 is homeomorphic to its own countable power. Using this fact twice, we conclude that Cp(X, D'0) contains a closed subspace homeomorphic to Cp(X, D) x w", tcontradicting

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the fact that the space C9(X,V) x w' is not Lindelof. Lemma 8.12 has been proved, and this also completes the proof of theorem 8.7.

IV.8.13. Corollary (E. A. Reznichenko). Assume MA + -iCH, and let X be a compactum for which (Cp(X))" is a normal space. Then X is a Frechet-Urysohn space, and satisfies the first axiom of countability on an everywhere dense set of points.

Proof. By theorem 8.7, the compactum X is Moreover, since CD(X) is a Lindelof space by Baturov's theorem (see chapts. I-III), the tightness of X is countable. Hence (see chapt. III, §3), X is a Frechet-Urysohn compactum, and the set of points at which X satisfies the first axiom of countability is everywhere dense in X. Bo-monolithic.

IV.8.14. Corollary. Assume MA+-'CH, and let X he a topologically homogencous compactum for which (CI,(X))°' is a normal space. Then X satisfies (at all points) the first axiom of countability, and JXJ < 2H0.

Proof. This follows from corollary 8.13 and the theorem about the cardinality of a compactum with the first axiom of countability (see 111.3.8).

IV.8.15. Corollary. Assume MA+-,CH, and let X be a compactum whose Suslin number is countable and for which (Cp(X))" is a normal space. Then X is metrizable.

Proof. By Baturov's theorem, the space (CD(X))" is a Lindelof space. Hence the tightness of X is countable (see chapt. 1, §4). By a theorem of B. E. Shapirovskii, c(X) < No and MA+--CH now imply that X is separable. Using MA+-'CH again and invoking theorem 8.7, we conclude that X is 8o-monolithic. But a separable Bo-monolithic compactum is metrizable.

Remark. As has been shown by E. A. Reznichenko, assertions 8.7, 8.13-8.15 remain true if the assumption of normality of (C,(X))" is replaced by the formally weaker requirement of normality of C7(X). The proof of this fact is somewhat more complicated. However, in the case of zero-dimensional compacta it is not difficult to obtain this result to full extent. Indeed, by theorem 8.6, if X is a zero-dimensional compactum and C,(X) is normal, then (CC(X))" is Lindelof. Hence we have

IV.8.16. Theorem (E. A. Reznichenko).

Assume MA + -CH. Then every

zero-dimensional compactum X for which CC(X) is normal, is lto-monolithic.

In a similar manner we can state analogs of assertions 8.13-8.15. Every Eberlein compactum is a continuous image of a zero-dimensional Eberlein compactum, every Corson compactum is a continuous image of a zero-dimensional Corson compactum, and every Gul'ko (Talagrand) compactum is a continuous image of a zero-dimensional Gul'ko (Talagrand) compactum. In relation to this, as well as

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to certain nuances arising in the above reasonings, it is natural to pose the following question.

IV.8.17. Problem.

Is it true that every compactum X for which C,(X) is a

Lindelof space is a continuous image of a zero-dimensional compactum X0 such that C,,(Xo) is a Lindelof space?

The similar question is also natural in the class of all Tikhonov spaces. The following problem, brought about by problem 8.17, is also useful.

IV.8.18. Problem.

What spaces Y can be represented as continuous images of spaces X for which CC(X) is a Lindelof space?

9. Lindelof E-spaces, and properties of the spaces Cr,,,,(X) In the sequence of spaces Cp(X),CC,(X),...,Cp,,,(X),C,,,,,II(X),... each snbsequent term is itch bigger than the previous one; moreover, looking at the terms of this sequence, we get the feeling that with increasing n the spaces CI,,,,(X) become more complicated and become spaces of more-and-more general nature. E.g., while C,(X) can (although seldom) have the Baire property or be Ncli complete, C,C,(X) has one of these properties only when X is empty, and CCCp(X) never has the Baire property. Only when X is empty will CPCP(X) be o-compact. However, the thesis about the increasing complexity of the spaces C,,,,(X) with increasing n should not be taken litterally. The symmetric duality theorems give us examples of `evenly stable' properties; among these are separability, monolithicity, stability. If a space X has any of these properties, then this property is present in all spaces Cp,,,(X) for even n E N+, but for n uneven the space CC,,,(X) need not have it. Sometimes we encounter absolute stability: e.g., if X is a space with a countable network, then all CC,,,(X) are spaces with a countable network; the Suslin numbers of all spaces CC,,,(X) are countable.

Nevertheless, within certain limits the thesis about the increasing complexity of the spaces Cp,,,(X) with increasing n is obviously true. In particular, experience and intuition tell us that with increasing n the spaces C,,,,(X) become less compact (as a matter of fact, already C,,(X) can have only very weak compactness type properties, and, as already noted, for X nonempty CCC(X) does not have the Baire property and is not o-compact). Among the very weak compactness properties we may reckon the Lindelof property. In relation to this there arises the following natural question: when are all spaces CC,,,(X) Lindelof spaces?

Even closer to compactness is the property of being a Lindelof E-space: it is a direct generalization of or-compactness. The following special version of the question above is of particular interest: when are all spaces CC,,,(X) Lindelof E-spaces? This question lies at the center of this section.

Note that if X is a space with a countable network, then all spaces Cp,,,(X) are

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Lindelof E-spaces. Is the converse true? This question has been open for quite some time, and only recently received a negative answer (0. V. Sipacheva). The following characterizations of Lindelof E-spaces are well known [66], [123].

IV.9.1. Proposition.

A space, X is a Lindelof E-space if and only if there are a countable family S of sets in X and a cover y of X by compact sets, such that if 4' E 'y and U is a neighborhood of the rompactum 4) in X, then there is a P E S for

which 4'CPCU.

IV.9.2. Proposition.

A space X is a Lindelof E-space if and only if in some (hence in any) compactification bX of X there is a countable family.F of compacta such that for each point x E X and for each point y E bX \ X there is a B E .F for which x E B and y V B. It is not difficult to prove directly that the conditions figuring in propositions 9.1 and 9.2 are equivalent.. In the sequel, a key role will be played by the following result of V. V. Uspenskii.

IV.9.3. Proposition. Let X be a Lindelof E-space. Then there is a Lindelof E-space Z such that CD(X) C Z C Rx. Proof. Let S = {Pn: n E N+} and -y be as in proposition 9.1. A realvalued (not necessarily continuous) function f on X is called S-bounded at a point x E X if there is a Pn E S such that x E Pn and f (Pn) is an R-hounded set. If a function f : X -+ R is S-bounded at all x E X, it is called S-bounded. Every realvalued continuous function on X is S-bounded. In fact, for an arbitrary x E X we can take a compactum 4' E ry such that x E 4'. Since f is continuous, it is bounded on some neighborhood U of 4). Choose now a P E S such that 4) C P C U. Clearly x E P, and f is bounded on P. Thus, for the subspace Z of Rx consisting of all S-bounded functions on X we have

C,,(X) C Z C Rx. We show that Z is a Lindelof E-space. Let R = R U {-co, oo} be the natural compactification of R homeornorphic to an

interval. Then RX is a compactum, and Cc(X) C Z C RX. In Particular, Rx is a compactification of Z. For arbitrary n, k E N+ we put RX Fn,k = {9 E : 9(Pn) C [-k, k]}.

Clearly, the set Fn,k is closed in R.X, hence compact. The family S = { FR,k: n, k E

N+} is countable. Take arbitrary f E Z and g E RX \ Z. We verify that there are n, k E N+ such that f E Fn,k and g 0 Fn,k. By proposition 9.2 this implies that Z is a Lindelof E-space. Since g Z there is a point xo E X at which g is not S-bounded. On the other

hand, f E Z implies that f is S-bounded at xo. Hence there are no, ko E N+ such

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that xo E Pnu and f (P, ) C [-ko, ko]. Then f E Fno,A,,. But g is not bounded on Pna, since it is not S-bounded at xo Hence g 0 F,,ko. Proposition 9.3 has been proved. Proposition 9.3 can be generalized somewhat.

IV.9.4. Proposition. If the Hewitt-Nachbin realcompactification vX of a space X is a Lindelof E-space, then there is a Lindelof E-space Z such that Cp(X)CZCRX. Proof. By proposition 9.3 there is a Lindelof E-space Z1 such that CC(vX) C ZI C R°X. By the restriction map 7r, the space R°X can be continuously projected onto RX. Moreover, by the definition of vX we have 7r(Cp(vX)) = Cp(X). Thus, Cp(X) = 7r(Cp(vX)) C 7r(ZI) C 7r(Wx) = Rx The space Z = 7r(Z1) is a Lindelof E-space, being a continuous image of the Lindelof E-space Z1. Proposition 9.4 has been proved.

IV.9.5. Theorem. If vCC(X) is a Lindelof E-space, then so is vX. Proof. By proposition 9.4 there is a Lindelof E-space Z such that CpCC(X) C Z C Rx. The space X is homeomorphic to the subspace X' = ti(X) of CCC(X), where b: X CpCp(X) is the evaluation map (see chapt. 0). Denote by Y the closure of X' in Z. Then Y is a Lindelof E-space. Moreover, Y is Hewitt-Nachbin complete. However, X' is C-embedded in RCP(x) (see chapt. 0), and every realvalued continuous function on X' can be extended to a realvalued continuous

function on Y. Hence Y = vX', and the space vX, which is homeomorphic to vX', is a Lindelof E-space. Theorem 9.5 is a mild generalization of the result of O. G. Okunev stating that vX is a Lindelof E-space if C,(X) is a Lindelof E-space. An advantage of our formulation of theorem 9.5 are its direct consequences:

IV.9.6. Corollary. If for some n E N+ the space vCp,,,(X) is a Lindelof E-space, then vX and vCp.,,,(X) are Lindelof E-spaces for all m < n.

IV.9.7. Corollary.

If a space X is R-complete (i.e. Hewitt-Nachbin complete),

while for some- n E N+ the space vCC,n(X) is a Lindelof E-space, then X is a Lindelof E-space.

The proof of the following theorem is a nice illustration of the use of theorem 9.5.

IV.9.8. Theorem.

If for some n E N+ the space vCp,n(X) is a Lindelof E-space, then X is loo-monolithic.

Proof. By theorem 9.5 it suffices to prove that if vCp(X) is a Lindelof E-space, then X is Ito-monolithic. But we know that if vY is stable then Y is l o-monolithic

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(chapt. 11, §6). Since every Lindelof E-space is stable, we conclude: if vCp(X) is a Lindelof E-space, then CC(X) is Ho-stable and hence X is Ho-monolithic (by results in chapt. II, §6).

IV.9.9. Corollary. If a space X is separable but does not have a countable network, then vCp,,,(X) is not a Lindelof E-space for any n E N+.

In the sequel we will need the following proposition, which can be regarded as a version of Grothendieck's theorem (see chapt. III, §4).

IV.9.10. Proposition. If vY is a Lindelof E-space, then every countably compact subspace F of Cp(Y) is a Gul'ko compactum, i.e. Cp(F) is a Lindelof E-space.

Proof. We canonically identify the points of the spaces Cc(Y) and Cp(W). This allows its to regard the set F as lying in Cp(vY). Since the topologies indnced by Cp(Y) and Cp(vY) on countable subsets coincide (see chapt. IL, §4), the set F is countably compact when regarded as a subspace of Cp(vY). However, vY is a Lindelof

E-space. Hence, by the result of D. P. Baturov (chapt. III, §6), F is a compactum when endowed with the topology induced by Cc(vY). But then the topologies induced on F by Cp(Y) and C7(vY) coincide. On the other hand, since vY is a Lindelof E-space and F is a compactum in Cp(v}'), the space Cp(F) is a Lindelof E-space, i.e. F is a Gul'ko compactum (§2). It is now easy to prove one of the basic results in the section.

IV.9.11. Theorem.

If F is a countably compact subspace of a space X and if for some n E N+ the space vCp,n(X) is a Lindelof E-space, then F is a compactum and Cp(F) is a Lindelof E-space (i.e., F is a Gul'ko compactum).

Proof. By theorem 9.5, vCp(X) is a Lindelof E-space. Put Y = CC(X). The space F is homeomorphic to a countably compact subspace F' of CCp(X) = CP(Y). Since vY = vCp(X) is a Lindelof E-space, we can use proposition 9.10. We conclude that F' and the space F homeomorphic to it are Gul'ko compacta.

IV.9.12. Corollary. If X is a countably compact space and vCpCp(X) is a Lindelof E-space, then X is a Gul'ko compactum. Theorem 9.11 and corollary 9.12 are mild generalizations of the result of 0. G. Okun-

ev stating that every compactum X for which CCC(X) is a Lindelof E-space, is a Gul'ko compactum. This result of 0. G. Okunev can also be understood as aparticular case of the following theorem.

IV.9.13. Theorem. If the space vCp(Y) is a Lindelof E-space, then every countably compact subspace F of Cp(Y) is a Gul'ko compactum.

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Proof. By theorem 9.5, vY is a Lindelof E-space. It remains to refer to proposition 9.10.

We will now prove another result in this section belonging to 0. G. Okunev, and going, with respect to theorem 9.11 and corollary 9.12, in the opposite direction.

IV.9.14. Theorem. Let Y C Cp(X), where X and Y are Lindelof E-spaces. Then CC(Y) is also a Lindelof E-space.

We first prove a somewhat simpler assertion.

IV.9.15. Proposition.

Let Y C Cp(X ), where X and Y are Lindelof E-spaces. Then Cp(Y, I) is also a Lindelof E-space (where I = [0, 11 is an interval).

Proof. Fix arbitrary e > 0, b > 0, n E N+, and P C Y. Denote by M(e, b, n, P) the subspace of the product IY x X' formed by all points (0, x1, ... , x") such that the values of 0 on two arbitrary functions f E Y and g E P which differ (in absolute value) at the points x1, ... , x by at most 6, are at most e apart. We verify that the set M(e, S, n, P) is closed in JY X X. Let (q', xi, ... , x;,) E M(e, b, n, P). Take arbitrary f E Y and g E P for which If (x;) - g(x;)( < d for all i = 1, ... , n, and assume that I0'(f) - V(g) I > e. By the definition of the topology in IY there is a neighborhood U of ¢' in IY such that 10(f) - 0(g)l > e for all 0 E U. Moreover, since f and g are continuous on X, there are neighborhoods Vi,..., V of the points x ... , x;, in X such that if (xi) -g(xi)l < b as soon as xi E V, i = 1, ... , n. Since (,0', x'1, ... , x,,) E M(e, b, n, P), there is a tuple (0, X1,..., x") E M(e, b, n, P) for which 0 E U and xi E V for i = 1... , n. But then If (xi) - g(xi) I < b for i = 1... , n, and by the definition of the set M(e, b, n, P) we have I q5(f) - 0(g) l < e. We have obtained a contradiction. Thus the closedness of M(e, b, n, P) in IY X X" has been proved. We now denote by B(e, 6, n, P) the image of M(e, b, n, P) under the projection of the space IY X X" onto the first factor. Thus, the subspace B(e, b, n, P) of IY consists of all functions -0 E IY for which there is a point (xi .... x,, ) of X" such that E M(e, b, n, P). (q5,x1,... , The space IY X X" is a Lindelof E-space, being the product of the Lindelof E-space

X' by the compactum IY. But then M(e, 6, n, P) is also a Lindelof E-space, being a closed subspace of IY x X", and B(c, 6, n, P) is also a Lindelof E-space, being a continuous image of M(e, b, n, P). Since Y is a Lindelof E-space, there are a cover -y of Y by compacta and a countable family S of subsets of Y such that for any 1 E -y and any neighborhood U of in Y there is a P E S for which 4> C P C U.

Further, let Q+ denote the set of positive rational numbers. We show that the countable family £ = {B(e, 6, n, P): e, b E Q+, n E N+, P E S) has the following property: *) whatever q5 E Cp(Y, I) and 0 E IY \ Cp(Y, I), there is a B(e, b, n, P) E £ such that ' E B(e, b, n, P) and z/i V B(e, 6, n, P).

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Since ip is discontinuous, there are g E Y and co E Q+ such that for arbitrary 6 > 0

and points xI,... , x E X there is a function f E Y for which If (xi) - g(xi)I < 6 for all i = 1,...,n, while I?G(f) - V)(g)I > eo Fix such a g and co. Choose a compactum 4'o e -y for which g E 4'o.

We introduce the following notation: if A C Y and xI, ... , x E X, 6 > 0, then V (A; x1,. .. , x,,; 6) denotes the set of all functions f E Y for which there is a function

h E A such that If (xi) - h(..Ti) I < 6 for i = 1,... , n. The set V (A; xi,... , x,,; 6) is open in Y and contains A. Since the function 0 E Cp(Y, I) is continuous on Y and 4'o is a compactum, the definition of the topology of pointwise convergence and its E X and a bo E Q+ such uniformity on the space imply that there are that if f E Y, h E 4'o, and If (xi) - h(xi)I < 26o, then Iq5(f) - q5(h)I < co. The set U = V ((Do; xI,... , x,,,; bo) is a neighborhood of 4'o in Y. Hence there is a Po E S for which 4'o C Po c lI = Take arbitrary g' E Po and f E Y such that If (xi) - g1(xi)I < 60 for i = I_. , no.

By the definition of U and the fact that Po C U, there is a III E 4'o for which Ig(ri) - h1(xi)I < ho for i I,...,no. Then If(.ri) - hl(xi)I <_ If(ri) - 9101'1)1 I

I91(xi) - hl (xi) I < bo + 5o = 26o for i = 1,... , no. and the number bo, we have I¢(f) - ¢(h1)I By the choice of the points x1,.. . , co. This means that (¢; XI, ... , x,,,,) E M(eo, bo, no, Po). Hence ¢ E B(eo, b0, no, Po)

Now g E 4'0 C Po and the choice of g and co imply that (V ; xi, ... , x,b) 0 E X. Hence V, 0 B(eo, 60, no, PO). M(eo, b0, no, Po), for any set of points x1, ... , Property *) of £ has been proved. Since E is a countable family and all its elements are Lindelof E-spaces, the following assertion completes the proof of proposition IV.9.15.

IV.9.16. Proposition (0. G. Okunev). Let bX be a compactification of a space X, and letF be a countable family of subspaces of bX such that 1) every B E .F is a Lindelof E-space;

2) for each pair of points x E X and y E bX \ X there is a B E F for which

XEBandy0B.

Then X is a Lindelof E-space.

Proof. For each B E F its closure B in bX is a compactification of B. Since B is a Lindelof E-space, there is, by proposition 9.2, a countable family SB of compacta in

B such that ifs

B and yE B\ B, then xE K andy K for some K ES. The

family S = U{SB: B E.F} U {B: B E F} is countable and consists of compacta.

We show that whatever x E X and y E bX \ X, there is a K E S for which x E K and y V K. By proposition 9.2 this means that X is a Lindelof E-space. By condition 2) there is a Bo E F such that x E Bo and y §t` Bo. If y ¢ Bo, then K = Bo is the required element in S. Suppose y E Bo. Then y e Bo \ Bo, and SB,, contains an element K such that x E K and y gt` K. Clearly, K is the required element in S. Propositions 9.15 and 9.16 have been proved.

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IV.9.17. Proposition.

If vY and Cp(Y, I) are Lindelof E-spaces, then CC(Y) is

also a Lindelof E-space.

Proof. By proposition 9.4 there is a Lindelof E-space Z such that Cp(Y) C Z C RY. Denote by R the canonical closure of R homeomorphic to the interval I: R = R U {-oo, oo}. Then Cp(Y, R.) is a Lindelof E-space by requirement. The spaces CC(Y), RY, and Cp(Y,R) are subspaces of RY. We have

Cp(Y)cZnCp(Y,R)cRYnCp(Y,R)=Cp(Y,R), whence C,(Y) = Z n CC (Y, R). Being the intersection of two Lindelof E-spaces lying in a common ambient space, Cp(Y) is itself a Lindelof E-space.

Proof of theorem 9.14. By proposition 9.15, Cp(Y, I) is a Lindelof E-space. Proposition 9.17 now implies that CC(Y) is also a Lindelof E-space.

IV.9.18. Corollary.

If X and Cp(X) are Lindelof E-spaces, then all Cp.n(X), where n E N+, are Lindelof E-spaces. If X is an Eberlein compactum, then Cp(X) is K-analytic (see chapt. IV, §2) and, hence, a Lindelof E-space. Hence corollary 9.18 implies

IV.9.19. Corollary (0. V. Sipacheva).

If X is an Eberlein compactum, then the spaces Cp,,,(X) are Lindelof E-spaces for all n E N+.

Recall that a compactum X is called a Gul'ko compactum if Cp(X) is a Lindelof E-space.

IV.9.20. Theorem. For every countably compact space X the following assertions are equivalent: a) there is an n E N+ such that vCC,n(X) is a Lindelof E-space;

b) X is a Gul'ko compactum; C) Cp,n(X) is a Lindelof E-space for every n E W.

Proof. By theorem 9.11, a) implies b). Let b) hold, i.e. X is a compactum and CC(X) is a Lindelof E-space. By corollary 9.18, all Cp,n(X) are Lindelof E-spaces, i.e. c) holds. Clearly, c) implies a).

By analogy with assertion 9.18 it is natural to suppose that if X and Cp(X) are K-analytic spaces, then the space CpCp(X) is also K-analytic. In this case, for every Eberlein compactum X all spaces Cp,n(X) would be K-analytic, which would give an analog of corollary 9.19. However, this hypothesis is not only unjustified, but even completely beyond the truth.

IV.9.21. Theorem (A. V. Arkhangel'skii). If the space CpCp(X) is K-analytic,

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then X is finite. We first show

IV.9.22. Proposition. Suppose CCp(X) is K-analytic. Then: a) every closed subspace Z of X having a countable network is finite; b) every convergent sequence {xn: n E N} in X is trivial; c) every compactum F in X is finite. Proof. a). The space X is itself K-analytic, being homeomorphic to a closed subspace of CpCC(X). Hence X is a Lindelof space; moreover, X is normal. Therefore the space

Cp(Z) is the image of Cp(X) under the continuous open restriction map, and the space CPCp(Z) is homeomorphic to a closed subspace of CDCp(X). Hence CPCP(Z) is K-analytic. But Z, Cp(Z), and CpCp(Z) are spaces with a countable network (see

chapt. I, §1). Hence CPC9(Z) is analytic [811. This implies (see chapt. 1, §1) that Cp(Z) is o-compact, and that Z is finite. b). Clearly, b) follows from a). c). Since CpCp(X) is K-analytic, it is a Lindelof E-space. By corollary 9.12, this implies that F is a Gul'ko compactum. Moreover, F is a FYechet-Urysohn space. But by b) F does not contain nontrivial convergent sequences. Hence the compactum F is finite. Proof of theorem 9.21. By proposition 9.21c), every compactum in X is finite. More-

over, X is a Lindelof E-space. But every Lindelof E-space in which all compacts are finite is countable (assertion IV.6.15). Hence X is countable. Applying proposition 9.22a) we conclude that X is finite. In this section, an important role has been played by proposition 9.4. In relation with it we pose the following question.

IV.9.23. Problem. Suppose that the space CC(X) (the space vC9(X)) is Lindelof. Is it then true that vX is Lindelof?

10. The Lindelof number of a function space over a linearly ordered compactum The main result in this short section is the following result of L. B. Nakhmanson 1361.

IV.10.1. Theorem.

If X is a linearly ordered compactum, then the Lindelof

number of Cp(X) is equal to the weight of X.

Proof. We have 1(Cp(X)) < mr(C9(X)) = nw(X) < w(X) for any space X (chapt. I, §1). =r.

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We show that for an arbitrary linearly ordered compactum X the opposite inequality holds:

w(X) < l(Cn(X)) Consider-the countable case: let CC(X) be a Lindelof space (in the general case only the terminology differs). We must prove that the compactum X is metrizable. We first prove that NI is a caliber of X. Assume the contrary: suppose there is a point-countable but not countable family

of nonempty intervals ((a0,b0): a < wI} in the space X with linear order < (of course, it is assumed that the order < generates the topology of X). I by f.,(x) = 0 for all For each a < wI we define a continuous function f0: X x < a,,, and f0(x) = 1 for all x > ba,. This is possible since X is normal. We show that the uncountable set { f,,: a < wI } does not have a complete accumulation point in CC(X), contradicting the fact that CC(X) is Lindelof. Let g E C,,(X). If at a certain point x' E X the function g takes a value g(x') not equal to 0 or 1, then the neighborhood 0Q = {f E Cp(X ): f (.T.') / 0, f t I) contains only countably many functions fp. In fact, fp E 09 implies that a, < x'_ < b0, i.e. x' E (a., b0). But, by requirement, the set {a < WI: x' E (aa b0)} is countable. It remains to consider the case when g(X) C (0,1 }. Since X is a compactum, (X, <) .contains the smallest point a'; at it all functions fa, vanish. (X, <) also contains the largest point 6'; at it all f,,, are equal to 1. Therefore g(a') = 0 and g(b') = 1. Thus

9'(1)0.

Using tlie.compactness of X, we can take b' = min g-'(1). We have a' < b'; denote

by a' the largest el'einent of the closed nonempty set {a E X: a < b'} = {a E X: a < b'} fl g-1 (0). Then g(a') = 0, g(b') = 0, and the interval (a', b') is empty, i.e. a' immediately preceeds b'.

Consider the neighborhood O9 = (f E Cp(X): f (a') < 1 and f (b') > 0} of g in Cp(X). Let f,,, E O. Then f0(a') < 1 and f0(b') > 0, whence a' < b0 and b' > a,,,. The relations a' < a, and b' >_ ba, cannot hold simultaneously, since the interval (a', b') is empty. Hence at least one of the points a', b' belongs to the interval (an, b,,). But there are only countably many such intervals (a0, b0). We conclude that the set {a < WI: fc. E O,} is countable. This proves that RI is a caliber of X. The tightness of X is countable, since C9(X) is a Lindelof space (chapt. 1, §4). But a linearly ordered space of countable tightness satisfies the first axiom of countability (see [661). Since every compactum satisfying the first axiom of countability and having caliber RI is separable [3], we conclude that X is separable. To complete the proof of theorem 10.1, we invoke some general considerations about linean) ordered spaces. I;et (Y, <) be a linearly ordered space. A pair of points a, b E Y is called a jump in (Y, <) if the interval (a, b) is empty, i.e. a immediately precedes b in (Y, <). A cut (A, B) in.-W., <) is any partition of Y into disjoint sets A and B such that a < b for all a E A, b e'B If.She. sets A and B are both nonempty, the cut is nontrivial. A cut (A, B) is called jumplike if A contains a largest element a and B contains a smallest element b; in this case the pair a, b is a jump in (Y, <). Conversely, if the pair a, b is

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183

a jump in (Y, <), then the sets A. = {y E Y: y < a} and Bb = {y E Y: b < y} form the jumplike cut (Aa, Bb) in (Y, <).

We relate to each cut S = (A,B) the function f(A,R): Y {0, 11 which is the characteristic function of B, i.e. is such that f(A,n)(A) C {0} and f(A,B)(B) C {1}. If this function f(A,B) is continuous with respect to the topology of Y generated by the order < and the discrete topology on {0,1 }, the cut (A, B) is called continuous. Clearly, the trivial cuts (0, Y) and (Y, 0) are continuous. Clearly, a nontrivial cut is continuous if and only if one of the two following conditions is satisfied: 1) the cut (A, B) is jumplike; 2) A does not contain a largest element and B does not contain a smallest element. Thus, the continuous cuts are precisely the `jumps' and `interior' gaps in the sense of R. Engelking [661. A function f : Y R. defined on a linearly ordered space (Y, <) is called a cutting

function if it, is continuous, f (Y) C {0, 11, and 7/, y" E Y, y' < y" implies f (y') < f (VI ). Clearly, the cutting functions on (Y, <) are precisely the functions of the forth f(A,B) where (A, B) is a continuous cut of (Y, <). The following assertions are obvious.

IV.10.2. Proposition.

The set S of all cutting functions on a linearly ordered space (Y, <) is closed in C(Y ).

IV.10.3. Proposition.

If a linearly ordered space (Y, <) is compact, then every nontrivial continuous cut of (Y, <) is jumplike. We also need the following general

IV.10.4. Proposition. Let (}, <) be a linearly ordered space, and let Z be the set of all cutting functions corresponding to jumplike cuts of (Y, <). Then all f E I are isolated in S, the space of all cutting functions on (Y, <); in particular, I is a discrete subspace of C(Y).

Proof. Let f E Z. There are a j. b1 E Y such that the interval (a1, b1) is empty while f(y) = 0 for all y < of and f(y) = 1 for all y > b1. C (}"): g(of) < 1 and 01 off does not contain any cutting function

10.2_!0.1 i. 1v

r V.10.5. Proposition. I= }'. <' 0 > +-' . . " .. 11 (.4.B i. c

..

...._.

_

.

:.. _ _

_.

.

..

Proof. By proposition 10.3, the set S of all cutting functions on (Y, <) has the form S = T U { fo, f, 1, where fo(Y) == {0} and f, (Y) = {I}. Since Y is compact, (Y, <) contains a smallest element a' and a largest element b'. Clearly f (a') = 0 and f (b') = 1 for all f E Z. Hence fo, fT 0 IT, and proposition 10.2

184

IV. LINDELOF NUMBER TYPE PROPERTIES FOR FUNCTION SPACES

implies that I is closed in CC(Y). On the other hand, by proposition 10.4 I is a discrete subspace of C,,(Y).

IV.10.6. Proposition. If a linearly ordered space (Y, <) is compact and CC(Y) is a Lindelof space, then the set of all continuous cuts of Y is countable.

Proof. This follows from propositions 10.3 and 10.5, since every closed discrete subspace of a Lindelof space is countable. We continue with the proof of theorem 10.1. We have established that the linearly ordered compactum (X, <) is separable if CC(X) is Lindelof. By proposition 10.6 it now suffices to prove the following assertion.

IV.10.7. Proposition.

If a linearly ordered compactum (X, <) is separable and the set of jumps in (X, <) is countable, then X has a countable base.

Proof. Let M be a countable everywhere dense set in X, and let {(an, bn): n E N} be the set of all jumps in (X, <). Denote by a*, respectively b*, the smallest, respectively largest, element of (X, <). Put M = M U {an, bn: n E N} U Ja', b' }. As can be readily seen, the family {(a, b): a, b E M} U { (y, an]: n e N, Y E M, y < an } U {(bn, y):

n E N, y e M, bn < y} is a countable network of X. But then the compactum X also has a countable base. The proof of theorem 10.1 is complete. We can somewhat strengthen theorem 10.1. Let (Y, <) be a linearly ordered space. Denote now by CD (Y) the subspace of C,(Y) consisting of all continuous monotone increasing functions f : Y - R. Clearly, C, -(Y) is closed in Cp(Y). Therefore, if the space Cp(Y) is Lindelof, then so is the space Cp (Y). We have the following proposition.

IV.10.8. Proposition.

For any interval (a, b) in a linearly ordered space (Y, <) there is a continuous monotone function f : Y -+ R such that f (y) = 0 for all y < a and f (y) = 1 for ally > b, while 0 < f (y) < 1 for all y E (a, b).

A proof of proposition 10.8 can be given by repeating, with obvious changes, the proof of the well-known Urysohn lemma about functional separatedness of closed sets in normal spaces.

Repeating the proof of the theorem of M. Asanov (see chapt. I, §4), taking into account proposition 10.8, we arrive at the following result.

IV.10.9. Theorem.

If (Y, <) is a linearly ordered space and if the space CP (Y) of all realvalued continuous monotone increasing functions on Y is Lindelof, then the tightness of Y is countable. Combining proposition 10.8 and theorem 10.9 with the reasonings given in the proof of theorem 10.1, we obtain the following assertion, generalizing theorem 10.1.

11. THE CARDINAUTY OF LINDELOF SUBSPACES

185

IV.10.10. Theorem. If (X, <) is a linearly ordered space and if the space C, -(X) of all realvalued continuous monotone increasing functions on X is Lindelof, then X is metrizable.

Since for every Corson compactum X the space Cc(X) is Lindelof, we obtain the following corollary to theorem 10.1.

IV.10.11. Corollary.

Every linearly ordered Corson compactum (in particular, every linearly ordered Eberlein compactum) is metrizable.

IV.10.12. Example.

As is well known, there exists a nonmetrizable linearly

ordered connected monolithic compactum X satisfying the first axiom of countability. The Aronshain continuum X is such a compactum. By theorem 10.1, the space C,(X) cannot he Lindelof. Thus, the theorem of E. A. Reznichenko, asserting that under the assumption of AMA + -CH every compactum X for which Cp(X) is a Lindelof space is tie-monolithic and has the Irechet-Urysohn property, has no converse.

The reasonings given in this section allow us to easily obtain the following result, which goes in a somewhat different direction.

IV.10.13. Theorem. If (X, <) is a linearly ordered space and the spread of CC(X) is countable, then X has a countable base.

Proof. We have s(X x X) < s(Cp(X)) < 1 o (see chapt. II, §5). Thus the Suslin number of X x X is countable. By P. Simon's theorem [31, if the Suslin number of the square of a linearly ordered space is countable, then this space is separable. Hence X is separable. By proposition 10.4, the cardinality of the set of jumps in (X, <) does not exceed the spread of Cp(X). Therefore the set of jumps of (X, <) is countable.

Fix a countable everywhere dense set A in (X, <). Put B = U{ {a, b}: (a, b) is a jump in (X, <)} U A. It is not difficult to verify that the countable family of sets of the form {x E X: x < c} and {x E X: d < x}, c, d E B, is a subbase of (X, <).

11. The cardinality of Lindelof subspaces of function spaces over compacts We have an exhaustive characterization of the compact subspaces of CC(X), with X an arbitrary compactum. They are precisely the Eberlein compacta. What can be said about the Lindelof subspaces of such CP(X)? In particular, for which compacta X does C,,(X) contain an everywhere dense Lindelof subspace? Of course, every subspace of Cp(X), with X a compactum, has countable tightness and is monolithic, since CC(X) itself has these properties. In this section we will first of all obtain bounds on the cardinalities of Lindelof subspaces of function spaces over compacta.

186

IV. LINDELOF NUMBER TYPE PROPERTIES FOR FUNCTION SPACES

We will also consider the dual problem: what can be said about the compacts in CC(X) if X is a Lindelof space?

IV.11.1. Theorem [77]. If X is a compactum and Y C CC(X), then IYI < 21ii1-`('

.

(1)

Before turning to the proof of theorem 11.1, we give two simple lemmas.

IV.11.2. Lemma. If X is a compactum, then for every countable set A C CP(X) we have

IAI <

2"0.

(2)

Proof. Since X is a compactum, the space C,,(X) is monolithic. Therefore A is a space with a countable network. But then IAI < 2"0.

IV.11.3. Lemma. If X is a compactum, then for every Y C CC(X) we have IYI < IYI"0.

(3)

Proof. Since X is a compactum, the tightness of C,(X) is countable. Hence,

Y= U{7: A C Y and IAI = No}. By lemma 11.2, IAI < 2140 for every countable set A C Y. We obtain

IYI < IYI"° .2"0 = IYI"°

Proof of theorem 11.1. Put T = 1(Y) c(X). It suffices to prove that IYI < 2T, since then, by lemma 11.3, IYI < IYI"0 = (2T)"0 = 2T. We have 1(Y) < T and t(Y) < t(Cp(X)) < No < T. Moreover, if A C Y and IAI = T, then IAI < T110 < 2T. If we prove that also ?P(Y) < 2', then by a well-known theorem about cardinalities (see [3], [16]) we have IYI < 2T.

Suppose that there is a point y' E Y for which V)(y', Y) > 2T. Then, as can be readily seen, l(Y \ {y'}) > 1G(y',Y) > 2T. Since X is a compactum, we may apply the theorem of D. P. Baturov (chapt. III, §6). We conclude: there is a closed discrete subspace A in Y \ {y'} such that IAI > 2T Put A' = A U {y'}. Then A' is closed in Y, and hence l(A') < T. Moreover A' contains only one nonisolated point, namely y'. Since l(A') < T, the cardinality of the complement A' \Oy of an arbitrary neighborhood Oy of y' in A' does not exceed T.

We may assume that y' is the function that vanishes identically on X. For each point x E X and e > 0 the set (f E Cp(X ): If (x)I < e} is a neighborhood of y', and hence I f f E Y: If (x)I > e}I < T. This implies that for every x E X,

I{f E Y : f(x) 0 0}I < T.

(4)

11. THE CARDINALITY OF LINDELOF SUBSPACES

187

Consider the diagonal product L1Y of the maps from Y, AY: X - RY. We denote by X' the subspace of RY that is the image of X under Y. Inequality (4) implies that X' lies in the E'-product of IYI copies of the real line R. Hence t(X') < r (see (3), [16]). Moreover, X' is a compactum, and c(X') < c(X) < r. By B. E. Shapirovskii's theorem,

w(X') < t(X')`(x*) < T' = 2. However, Y is homeomorphic to a subspace of Cp(X') (see chapt. 0). Therefore IYI < ICC(X')I < (d(CC(X'')))"0 (we have used lemma 11.3). But d(Cc(X')) < nw(CC(X')) = nw(X') = w(X*) < 2' (see chapt. I, §1). Hence (2")NO

IYI

= 2T.

Theorem 11.1 has been proved.

IV.11.4. Corollary.

If X is a compactum whose Suslin number is countable, and Y is a Lindclof subspace of Cc(X), then IYI < 2"0.

IV.11.5. Corollary.

If X is a coinpactuin with countable Suslin number, and Cp(X) contains an everywhere dense Lindelof subspace, then w(X) < ICp(X)I < 2"0.

Proof. We use corollary 11.4 and refer to the fact that w(X) = nw(X) = nw(Cp(X)) 5 ICp(X)I <_

2"0.

In particular, the last conclusions are valid for dyadic compacta, for the Suslin number of such compacts is countable. However, in this case the conclusion can be considerably strengthened.

IV.11.6. Theorem [771.

Let X be a dyadic compactum, and Y C Cp(X). Then

nw(Y) = 1(Y).

Proof. Consider the image X' of X under the diagonal product AY of the maps from Y. Under the evaluation map 2b: Y -> CC(X') the space is homeomorphically mapped

onto a subspace Y' of Cp(X*) separating the points of the colnpact.um X' (see chapt. 0). We have (see chapt. 0) nw(Y) = nw(Y*) < nw(Cp(X')) = nw(X') < w(X'). It remains to prove that w(X') < 1(Y). The compactum X' is dyadic, hence its weight is equal to the least upper bound of the cardinals r for which the space D' = {0,1}T can be topologically embedded in X' [66]. This implies that the weight of X' is equal to the least upper bound of the cardinals r for which T(r + 1) = {a an ordinal, a < r}, the space of all ordinals not exceeding the first ordinal r of cardinality r (in the usual topology), can be topologically embedded in X'. Now if the compactum T(r + 1) is homeomorphic'to a subspace of X', then the restriction map continuously maps Y' onto a subspace Z of Cp(T(r + 1)) separating the points of T(r + 1). Here, 1(Z) < l(Y') = l(Y). It remains to prove the following assertion.

IV. LINDELOF NUMBER TYPE PROPERTIES FOR FUNCTION SPACES

188

IV.11.7. Proposition.

Let r be a regular cardinal, and Z C CC(T(r + 1)), Z separating the points of T(r + 1). Then 1(Z) > r.

Proof. For an arbitrary a < r we fix a function fa E Y and rational numbers S., to such that one of the two following conditions is satisfied: 1) fa(a) < sor < to < fa(r); 2) f., (r) < to < Sa < fa(a). For each ordinal a > 0 there is an ordinal /3(a) < a such that if ry E (,Q(a), a], then

fa('Y) < sa < to in case 1), and to < sa < f,,(-t) in case 2). By Fodor's lemma there are a set E unbounded in {a: a an ordinal, a < r}, an ordinal 8 < r, and rational numbers s, t such that /j(a) = /3, sa = s, to = t for all a E E. Here we may assume that that 1) is satisfied for all a E E. Since r is regular, jEH = r.

We show that the set M = { f, : a E E} does not have a complete accumulation point in the space Cc(T(r + 1)).

Take an arbitrary function f' E CC(T(r + 1)) such that f' E IT. Since f(a) <

s
s < t < f'(r). Put s' = (s + t)/2. Since f' is continuous, there is an a' < r for which /3 < a' and

s' < f'(a').

Consider the neighborhood O. = {f E C,,(T(r+1)): s' < f (a')} of f' in CC(T(r+ M. We show that the membership fa E Of- can hold for only those a E E for which a < a*. By the regularity of r this will imply that we cannot extract a subcover of cardinality less than r from a cover of M by sets of the form Of.. So, let a E E and a' < a. But 3(a) = 0 < a'. Hence, by the definition of 13(a), fa(a') < s < s' < t. Therefore fa f Of.. The proof of proposition 11.7 and theorem 11.6 is complete.

IV.11.8. Corollary [77]. If X is a dyadic compactum, then every Lindelof subspace of CC(X) has a countable network.

IV.11.9. Corollary. If X is a nonmetrizable dyadic compactum, then CC(X) does not contain an everywhere dense Lindelof subspace.

Note that proposition 11.7 implies

IV.11.10. Corollary. The space Cp(T(w1+1)) does not contain a Lindelof subspace separating the points of the compactum T(w1 + 1).

In relation to this there arises a natural question: for which compacta X does Cp(X) contain a Lindelof subspace separating the points of X?

11. THE CARDINALITY OF LINDELOF SUI3SPACCS

189

IV.11.11. Problem.

Let. X be a compactum, and let C,,(X) contain a Lindelof subspace separating the points of X. Is then the tightness of X countable? What. if C,,(X) contains an everywhere dense Lindelof subspace separating the points of X? We show that if X is not assumed to be compact, then the answer to the question posed above is negative.

Let X = L(RI) be the one-point Lindelofication of the IV.11.12. Example. discrete space of cardinality R1. Then CP(X) is homeomorphic to the E-product of real lines, and contains an everywhere dense or-compact subspace. At the same time, X is a Lindelof P-space. Problem 11.11 can be dually formulated. We first agree on some terminology. We call a space Y supLindelof if there is a Lindelof space X such that Y is honieomorphic

to a subspace of C,(X). The following problem is easily seen to be equivalent to problem 11.11.

IV.11.12'. Problem.

Is it true that the tightness of every supLindelof compactum

is countable?

Note that if X is a Lindelof space such that X' is also Lindelof, for all n E N+, then every compactum in C,,(X) has countable tightness (see chapt. II, §1). The problems 11.11 and 11.12 would receive a negative answer if the following problem would be answered in the negative.

IV.11.13. Problem. Is it true that a continuous image of a supLindelof compactum is a supLindelof compactum?

We now show that a positive answer of problems 11.11 and 11.12 can be obtained under assumption of the proper forcing axiom PFA.

IV.11.14. Theorem.

Assume PFA, and let Y be a compactum, Y C C,(X),

where X is a Lindelof space. Then t(Y) = No. We first show the following proposition.

IV.11.15. Proposition. Let Y be a compactum, Y C Cp(X ), and Y of uncountable tightness. Then there is a closed subspace XI of X such that Cp(X1) contains a compactum of weight I21 and of uncountable tightness.

Proof. Since Y is compact and t(Y) > 1 o, there is a free sequence S = {yQ: a < wl} in Y of length I (see [3), [161). Y contains a complete accumulation point y' for S.

Put Sp = {ya: a < 6}, for 3 < wl. Then y' E S and y' f Sp for /3 < w1. By the definition of topology of pointwise convergence, for each /3 < wl there is a finite set

190

IV. LINDELOF NUMBER TYPE PROPERTIES FOR FUNCTION SPACES

ICp C X such that for the restriction map 7rp, sending a function f E Cc(X) to the function f 1K E C1,(Kp), we have

'roW) V ro(So) = {1p(ya) : a _< fi}.

Put A = U{Kp: 13 < wl } and XI = A. Consider also the restriction map 7r: CC(X)

Cp(X1), where 7r(f) = f Ixl for f E CC(X). The density of X1 does not exceed the cardinality of A, i.e. d(X1) = N. Hence the space Cp(X1) can be condensed onto a space of weight < NI (see chapt. 1, §1). Consequently, the weight of the ,

compact subspace Y1 = 7r(Y) of Cp(X1) does not exceed N1. Since 7r is continuous, 7r(y*) E 7r(S) = {7r(ya) : a < w1}. But since ICp C X1i for all,6 < w1 we have

lr(y`) 0 x(S,) = {7r(ya) : a <M. This implies that for every countable set B C 7r(S), 7r(y')

J. Now y' E Y and

S C Y imply that 7r(y`) E Y1 and 7r(S) C Y1. We conclude that the tightness of Y1 is uncountable and that the weight of Y1 is Ni.

Proof of theorem 11.14. Assume the contrary. Let the tightness of Y be uncountable.

By proposition 11.15, there is then a closed subspace X1 of X such that Cp(X1) contains a compactum of uncountable tightness and of weight N1.

As has been proved by Z. Balogh, PFA implies that any compactum of weight 81 and of uncountable tightness contains a compactum which is homeomorphic to the space T(w1 + 1) of all ordinals not exceeding the first uncountable ordinal. We conclude that Cp(X1) contains a compactum F which is homeomorphic to T(w1 + 1). The space X1 is a Lindelof space. Its image under the canonical evaluation map 1/F : X 1 - Cp(F) is a Lindelbf subspace of CC(F) separating the points of F (chapt. 0). However, this contradicts corollary 11.10, since F is homeomorphic to T(w1 + 1). PFA implies that every compactum of countable tightness is sequential (Z. Balogh). Combining this result and theorem 11.14, we obtain the following conclusion.

IV.11.16. Theorem.

Assume PFA. Then every supLindelof compactum is se-

quential.

IV.11.17. Problem. Is it also true that under the assumption MA + -NCH every supLindelof compactum is sequential? It is well known that MA+-,CH does not imply that every compactum of countable tightness is sequential (P. Nyikog).

IV.11.18. Problem.

Is it true that under the assumption MA + -iCH every

supLindelof compactum is No-monolithic? Let Y be a compactum in CC(X), and

let X" be a Lindelof space for every n E N+. Does MA + -CH imply that Y is No-monolithic? That it is sequential?

11. THE CARDINALITY OF LINDELOF SUBSPACES

191

The cardinality of every topologically homogeneous sequential compactum does not exceed 2"0 13], [16]. This and theorem 11.16 imply

IV.11.19. Theorem. Let X be a topologically homogeneous compactum, and suppose there exists a Lindelof subspace Y in CP(X) separating the points of X. Assuming

PFA, we have IXI < 2"0.

IV.11.20. Problem. Does theorem 11.19 remain true if we do not assume PFA? Note that up till now it is not known whether every topologically homogeneous compactum of countable tightness satisfies the fist axiom of countability (in ZFC!).

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Index

No-hounded space 36 adequate compacturn 157 adequate family of subsets 156 Aleksandrov supersequence 151 algebraic-topological properties 2

entier function 22 extent 5

Amir-Lindenstrauss theorem 150 analytic space 7, 48, 99 Arkhangel'skiT-Pytkeev theorem 45 Arkhangel'skiT-Reznichenko theorem 169 Arkhangel'skit theorem 62, 77, 78, 85, 100, 120, 130,

finitely separating set 73 fractional part 22

fan tightness 49

finitely generating set 73 R6chet-Urysohn fan 94 Fr6chet-Urysohn space 7, 51 functional generation 109 functional tightness 58 functionally closed map 15 functionally closed space 57 functionally perfect space 127

180

Asanov theorem 33 Asanov-Velichko theorem 106 Faire property 7, 31 Balogh theorem 190

generating family of maps 17

base 5

generating set of functions 73 Crothendieck theorem 107 Cul'ko compactum 177

Baturov theorem 121, 123 bounded subset 6

Cul'ko-Alster-Pol theorem 136 Cul'ko-Michael-Itudin theorem 143

C-embedded subspace 6 Calbrix theorem 100 caliber 10 canonical closed set 60 Cech complete space 7, 31 character 5

Hamel basis 21 Haydon theorem 113 hedgehog-like space 129

Hewitt-Nachbin complete space 6 Hewitt-Nachbin number 60 Hewitt-Nachbin space 57 Hurewicz space 48

Chigogidze theorem 61 cofinality 4 compact-open topology 8 completely screenable space 135 condensation 4 conjugate sets 144 continuous cut 183 continuously invariant property 109 convergent sequence of subsets 152 Corson compactum 134

i-weight 5

jump 182 jumplike cut 182 K-analytic spaces 7, 99

Corson theorem 41 countable set 4

k-directed class of topological spaces 125

countably centered family 35 countably pracompact space 108

k-primary Lindelof space 143

cozero set 151

k-space 6, 51 ke6-directed class of spaces 134

k-separable space 127

cut 182 cutting function 183

Korovin theorem 35, 36 Kowalsky theorem 130

D-separating family 97 density 5

l-equivalent spaces 8 left ray 68

derived set 166

diagonal 5

left space 68

diagonal number 5 Dieudonn6 complete space 6, 57 discrete Lindelof number 84

left well order 68 Lindelof E-space 6, 47 Lindelof p-space 80 Lindelof number 4 Lindelotication 4 linear topological property 2 local oscillation 118 lower limit of a sequence 51

dual 17 dually-k-separable spaces 127

Eberlein compactum 1, 102 Eberlein-Crothendieck space 95 EC-space 95 Engelking factorization lemma 81

p-space 6 203

204

INDEX

m,-space 60 map onto 4

Rosenthal theorem 120, 151

metaLindeli f space 135 minimal well ordered set 4 monolithic space 76 Moscow space 60 multiplicative functional 22

E-product 7 E.-product of real lines 149

Nagata theorem 22 Nakhmanson theorem 181 Namioka theorem 118

network 4 network weight 5

nonmetrizable countable hedgehog 27 Nyiko8 theorem 190

a-compact space 5 0-point-finite family of sets 151 a-space 35

S-bounded function 175 scattered space 7, 85 separating family of maps 16 sequential space 7, 51 set of type 0, 60 Simon theorem 185 simple elementary open set 73 simple space 84 Sipacheva theorem 180

w-cover 51 O-convex subspace 40 o-Lindel6f spaces 163

Okunev example 111 Okunev theorem 22, 65, 97, 100, 113, 177, 178, 179 P-space 6, 28, 47 P,-space 84

perfect map 6 perfectly-K-normal space 11 placement property 109

Pal example 164 Pol theorem 160 position property 109 precaliber 10 Preiss-Simon space 152

Preiss-Simon theorem 153 primary Lindeli f space 137 protectively complete space 87 property e 53 property of boundedness type 110 pseudocharacter 5 pseudocompact space 6 Q-space 57 Q,-space 60

R-bounded subset 6 R-complete space 6, 57 R-extent 36 R-quotient map 14 R-quotient topology 14 R-tightness 59 real quotient map 14 real quotient topology 14 realcompact space 57 realcomplete space 57

regular family of maps 16 resoluble compactum 147 resoluble family of retractions 147 Reznichenko theorem 38, 170, 173

space of polntwise countable type 6 space of type Ke6 7, 132 spread 5, 66 stable space 76 strictly r-continuous function 59 strong condensation property 135

strong functional generation Ill strongly r-monolithic space 83 strongly monolithic space 83 supertopological property 2 supLindel6f space 189 Suslin number 4

r-continuous function 58 r-hull 131 r-invariant space 131 r-monolithic space 76 r-perfect class of spaces 133 r-placed set 60

r-simple space 84 r-stable space 76 t-equivalent spaces 8 7b-separating family of sets 151 Talagrand example 159 Talagrand theorem 132 tightness 5 Tikhonov cube 4 Tikhonov spaces 4 Tkachuk example 128 Tkachuk theorem 37, 88 topologically homogeneous space 105 topology of pointwise convergence 8 topology of uniform convergence on the elements of the

set E8 u-equivalent spaces 8 u-properties 2 Ulam-measurable cardinal 6 UspenskiT theorem 62, 64, 86, 96, 175 Velichko theorem 28, 75

right ray 68 right space 68 right well order 68

weak functional tightness 59 weak tightness 58

ring properties 2

weakly Lindel6f space 163

INDEX weight 4

Z-set 6 Zenor-Velichko theorem 69 zero set 6

20'