THE INFINITE-DIMENSIONAL TOPOLOGY OF FUNCTION SPACES
North-Holland Mathematical Library Board of Honorary Editors: M. Artin, H. Bass, J. Eells, W. Feit, P.J. Freyd, F.W. Gehring, H. Halberstam, L.V. Hormander, J.H.B. Kemperman, W.A.J. Luxemburg, P.P. Peterson, I.M. Singer and A.C. Zaanen
Board of Advisory Editors: A. Bjorner, R.H. Dijkgraaf, A. Dimca, A.S. Dow, JJ. Duistermaat, E. Looijenga, J.P. May, I. Moerdijk, S.M. Mori, J.P. Palis, A. Schrijver, J. Sjostrand, J.H.M. Steenbrink, F. Takens and J. van Mill
VOLUME 64
ELSEVIER Amsterdam - London - New York - Oxford - Paris - Shannon - Tokyo
The Infinite-Dimensional Topology of Function Spaces
Jan van Mill Faculteit der Exacte Wetenschappen, Amsterdam, The Netherlands
2001 ELSEVIER Amsterdam - London - New York - Oxford - Paris - Shannon - Tokyo
ELSEVIER SCIENCE B.V. Sara Burgerhartstraat 25 P.O. Box 211, 1000 AE Amsterdam, The Netherlands © 2001 Elsevier Science B.V. All rights reserved. This work is protected under copyright by Elsevier Science, and the following terms and conditions apply to its use: Photocopying Single photocopies of single chapters may be made for personal use as allowed by national copyright laws. Permission of the Publisher and payment of a fee is required for all other photocopying, including multiple or systematic copying, copying for advertising or promotional purposes, resale, and all forms of document delivery. Special rates are available for educational institutions that wish to make photocopies for non-profit educational classroom use. Permissions may be sought directly from Elsevier Science Global Rights Department, PO Box 800, Oxford OX5 1DX, UK; phone: (+44) 1865 843830, fax: (+44) 1865 853333, e-mail:
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0 444 50557 1 0924-6509
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Contents Introduction
xi
Chapter 1.1. 1.2. 1.3. 1.4. 1.5. 1.6. 1.7. 1.8. 1.9. 1.10. 1.11.
1. Basic topology Linear spaces Extending continuous functions Function spaces The Borsuk homotopy extension theorem Topological characterization of some familiar spaces The inductive convergence criterion and applications Bing's shrinking criterion Isotopies Homogeneous zero-dimensional spaces Inverse limits Hyperspaces
1 1 21 29 37 41 58 66 70 73 80 95
Chapter 2.1. 2.2. 2.3. 2.4. 2.5.
2. Basic combinatorial topology Affine notions Barycenters and subdivisions The nerve of an open covering Simplices in Rn The Lusternik-Schnirelman-Borsuk theorem
Ill Ill 125 132 138 148
Chapter 3.1. 3.2. 3.3. 3.4. 3.5. 3.6. 3.7. 3.8. 3.9. 3.10. 3.11. 3.12.
3. Basic dimension theory The covering dimension Translation into open covers The imbedding theorem The inductive dimension functions ind and Ind Dimensional properties of compactifications Mappings into spheres Dimension of subsets of Rn and certain generalizations Higher-dimensional hereditarily indecomposable continua Totally disconnected spaces The origins of dimension theory The dimensional kernel of a space Colorings of maps
151 151 157 168 176 183 193 204 210 216 221 227 237
3.13. 3.14.
Various kinds of infinite-dimensionality The Brouwer fixed-point theorem revisited
251 257
Chapter 4.1. 4.2. 4.3.
4. Basic AMR theory Some properties of ANR's A characterization of ANR's and AR's Open subspaces of ANR's
263 263 277 301
Chapter 5.1. 5.2. 5.3. 5.4. 5.5.
5. Basic infinite-dimensional topology Z-sets Extending homeomorphisms in s The estimated homeomorphism extension theorem The compact absorption property Absorbing systems
307 307 311 320 329 343
Chapter 6.1. 6.2. 6.3. 6.4. 6.5. 6.6. 6.7. 6.8. 6.9. 6.10. 6.11. 6.12. 6.13.
6. Function spaces Notation The spaces CP(X): Introductory remarks The Borel complexity of function spaces The Baire property in function spaces Filters and the Baire property in CP(N?) Extenders The topological dual of CP(X) The support function Nonexistence of linear homeomorphisms Bounded functions Nonexistence of homeomorphisms Topological equivalence of certain function spaces Examples
367 368 369 372 377 387 393 399 404 411 416 426 434 445
Appendix A. Preliminaries A.I. Prerequisites and notation A.2. Separable metrizable topological spaces A.3. Limits of continuous functions A.4. Normality type properties A.5. Compactness type properties A.6. Completeness type properties A.7. A covering type property A.8. Extension type properties A.9. Wallman compactifications A.10. Connectivity A.11. The quotient topology A. 12. Homotopies A. 13. Borel and similar sets
457 457 465 468 469 473 479 485 490 494 500 505 510 517
Appendix B.
Answers to selected exercises
527
Appendix C.
Notes and comments
579
Bibliography
597
Special Symbols
613
Author Index
615
Subject Index
619
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Introduction In this book we study function spaces of low Borel complexity. This is a particularly interesting class of spaces; to investigate it one needs a mix of methods and techniques from areas as diverse as general topology, infinite-dimensional topology, functional analysis and descriptive set theory. A striking result is the theorem of Dobrowolski, Marciszewski and Mogilski, which states that all function spaces of low Borel complexity are topologically homeomorphic. A major feature of this book is a complete and self-contained proof of this remarkable fact. In order to understand these details, a solid background in infinitedimensional topology is needed. And for that one needs to know a fair amount of dimension theory as well as ANR theory. The necessary material was partially covered in my previous book 'Infinite-dimensional topology, prerequisites and introduction'. A selection of what was done in that volume can also be found here, but completely revised and in many places expanded with recent results. I chose to take a 'scenic' route towards the Dobrowolski-Marciszewski-Mogilski Theorem, that is linking the results needed for the theorem's proof to interesting recent research developments in dimension theory and infinite-dimensional topology. The first five chapters of this book are intended as a text for graduate courses in topology. For a course in dimension theory, one should cover Chapters 2 and 3 and part of Chapter I . For a course in infinite-dimensional topology, one should cover Chapters 1, 4 and 5. In Chapter 6, which deals with function spaces, I discuss recent research results. It can also be used for a graduate course in topology but its focus is more suited to that of a research monograph than of a textbook; it would, therefore, be more appropriate to use it as a text for a research seminar. This book, consequently, has the character of a textbook as well as a research monograph. In Chapters 1 through 5, unless stated otherwise, all spaces under discussion are separable and metrizable. In Chapter 6 results for more general classes of spaces are presented. In Appendix A we collected for easy reference and for sake of completeness some basic facts that are important in the book. The reader will see that it is not intended as the basis for a course in topology; its purpose is to collect what one should know about general topology, nothing more nothing less. The exercises in the book serve three purposes: to test the reader's understanding of the material, to supply proofs of statements that are used in the text and are not proven therein, and to provide additional information not covered by the text. We included the solutions to selected exercises in
Appendix B. These exercises are important or difficult; they are marked in the text by the symbol >•. If the reader wants to find the meaning of some unfamiliar term in this book, it is best to check Appendix A first, since many basic concepts are defined there. To simplify the search process, in the index all page numbers for terms from Appendix A are italicized. For example, if the reader would like to know the definition of the term 'topologically complete', she or he should look at Page 480. For in the index, the first italicized page number under 'topologically complete' is 480. Finally, I express my indebtedness to Jan Baars, Stoyu Barov, Jan Dijkstra, Tadek Dobrowolski, Klaas Pieter Hart, Michael van Hartskamp, Henryk Michalewski, Witek Marciszewski, Roman Pol, Ruud Salomon and Jan de Vries for their critical reading of parts of the manuscript and their many valuable suggestions for improvements. None of these distinguished colleagues is responsible for the remaining errors, which are mine.
Jan van Mill Bussum, March 29, 2001
CHAPTER 1
Basic topology In this chapter we present some basic facts on the topology of separable metrizable spaces. We discuss linear spaces, inverse limits, hyperspaces, Bing's Shrinking Criterion, etc. Questions about the possibility of extending continuous functions, or creating new continuous functions from old ones, are central in this chapter. Applications include a proof of the topological homogeneity of the Hilbert cube and proofs of topological characterizations of various interesting spaces such as the Cantor set, the unit interval and the spaces of rational and irrational numbers, respectively. Many of the results presented in this chapter are geometrically motivated, although this is not always clear at first sight. For background information, see Appendix A. Our conventions with respect to notation can be found in §A.l. All topological spaces under discussion are separable and metrizable. 1.1. Linear spaces
A linear space is a real vector space L carrying a (separable metrizable) topology with the property that the algebraic operations of addition and scalar multiplication are continuous (warning: a vector space is an algebraic structure which may or may not carry a topology while a linear space is automatically a topological space). Observe that the continuity of the algebraic operations on a linear space show that it is a topological group. A subset A of a linear space L is called convex if for all x, y £ A and a G I we have ax + (1 — a)y G A. Let L be a linear space. If A C L then conv(A) denotes the smallest convex subset of L containing A] this set is called the convex hull of A. A convex combination of elements of A is a vector of the form ^T=i ^ai with a i , . . . , on € A, AI , . . . , An € I and Y%=i ^i = 1 • For each n G N, let convn(A) denote the set of vectors x G L that can be written as a convex combination of at most n vectors from A. Also, put = [J conv n (A)
2
1. BASIC TOPOLOGY
and observe that convoo (A) is the set of all convex combinations of elements of A It is left as an exercise to the reader to present a proof of the following basic lemma (see Exercise 1.1.5). Lemma 1.1.1. Let L be a linear space with subset A. Then (1) conv(A) = (2) if A is finite then conv(A) is compact. A linear space L is called locally convex if the origin of L (which we shall always denote by 0) has arbitrarily small convex neighborhoods. Obviously, the spaces Rn for n G N U {00} with their usual product topologies are locally convex linear spaces under coordinatewise denned addition and scalar multiplication. Let L be a vector space. A norm on L is a function ||-||: L —> [0, oo) having the following properties: (1) (2) (3)
\\x + y\\ < \\x\\ + \\y\\ \\tx\\ = \t\ • \\x\\ \\x\\ = 0
f o r a l l z , y e L, for alU 6 M and x € L, if and only if x = 0.
If || -|| is a norm on L then the function g ( x , y ) = \\x-y\\ defines a metric on L; it is called the metric derived from the norm \\-\\. We call a linear space L normable provided that there exists a norm on it such that the metric derived from this norm is admissible; for obvious reasons such a norm is called admissible. Observe that each normable linear space is locally convex (See Exercise 1.1.6). A normed linear space is a pair (L,\ -||), where L is a vector space and ||-|| is a norm on L. We shall always endow the underlying vector space of a normed linear space with the topology derived from its norm. So we make a formal distinction between normed linear space and normable linear space: a normable linear space may possess many different norms that generate its topology, see Exercise 1.1.8, whereas in a normed linear space the norm is fixed. In topology we make a similar distinction between metric and metrizable spaces. A metrizable space may posses many admissible metrics generating the same topology, whereas in a metric space the metric under consideration is fixed. If (V, || -||) is a normed linear space then
B = {x 6 V : \\x\\ < 1} and
S = x£ B: x = 1
1.1. LINEAR SPACES
3
denote its unit ball and unit sphere, respectively. A subset A of a normed linear space L is called bounded if there exists an e E [0, oo) such that ||a|| < e for every a £ A. A Banach space is a normable linear space for which there exists an admissible norm such that the metric derived from it is complete. So a Banach space is topologically complete by definition, and hence is a Baire space by Theorem A.6.6. Examples of linear spaces. We will now discuss various important examples of linear spaces. Example 1.1.2. The Euclidean spaces W1. The standard norm on W1 is the Euclidean norm which is defined by
As noticed on Page 461, Bn and §n l abbreviate the unit ball and unit sphere in W1. It is clear that W1 is a Banach space since the metric derived from its Euclidean norm is the well-known Euclidean metric which is complete. Example 1.1.3. The space s. The space s is the vector space E°° endowed with the Tychonoff product topology. It is a classical object both in topology and functional analysis and will play a central role in the remaining part of this book. Observe that s is topologically complete by Lemma A.6.2. Its standard complete metric is the following one:
/
CO
I
\ _ V^ 2 _ n n=l
I
\xn-yn\ ~r I n
J/n|
(See Exercise 1.1.1 for the verification.) Since each Mn is normable and since s is in many respects their 'limit', the question naturally arises whether s is normable. We will show below that it is not. Define cr = {x € s : xn = 0 for all but finitely many n G N}. It is clear that cr is a linear subspace of s. Lemma 1.1.4. If L is a linear subspace of s with cr C L then L, endowed with the subspace topology it inherits from s, is not normable. Proof. Assume, to the contrary, that ||-|| is an admissible norm on L. Then U = {xeL: \\x\\ < 1}
1. BASIC TOPOLOGY
is an open neighborhood of the origin of L. By definition of the product topology on s there are an open neighborhood V of 0 in R and an n G N such that
(*)
x
(n^ n ^)n£cc/, n
i=l
oo
i=n+l
where Vi = V for i < n and Rj = IR for i > n. Let y 6 s be defined by yi = 0 if i ^ n + 1 and yn+\ = 1. Since y E a C L and y 7^ 0 it follows that e = ||y|| > 0. By (*), £ y 6 U for every t € E. In particular, ||s/e|| < 1 but also 1 1 Veil — £/e — 1> which is a contradiction. D From the proof of Lemma 1.1.4 it is clear that the interplay between the topology and the linear structure on s prevents it from being normable. (The question naturally arises whether every vector space can be endowed with a norm which is compatible with its linear structure. The answer to this question is in the affirmative, see Exercise 1.1.8.) Consequently, although s seems a natural 'limit' of the spaces IRn, it is notably different from any of its finite dimensional analogs W1 . Example 1.1.5. The spaces C(X) and C*(X). For a nonempty compact space X we let C ( X ) denote the set of all continuous real-valued functions on X. Obviously, C(X) is a vector space; addition of functions and scalar multiplication are defined pointwise. If / e C(X) then define its norm, ||/||, by H/ll =
sup{\f(x)\:xeX}.
(Observe that by compactness of X this supremum is attained. That is: there is an element x e X such that ||/|| = \f(x}\.) It is easily seen that ||-|| : C(X] —> [0, oo) is indeed a norm; it is called the sup-norm on C(X). Consequently, the function (*)
Q(f, 9) = \\f~9\\
defines a metric on C(X) and therefore generates a topology. From now on we shall endow C(X] with this topology. There are other useful and interesting topologies on C(X). In Chapter 6 we shall endow C(X) with the so-called topology of pointwise convergence. It will be clear from our notation which topology on C(X] we are using. For example, C(X) denotes the set C(X] endowed with the above topology, and CP(X) denotes the set C(X) endowed with the topology of pointwise convergence, etc. We claim that C(X) is a Banach space. Let ( f n } n be a £-Cauchy sequence. Then ( f n } n clearly converges pointwise, so / = limn^.^ fn exists and belongs to C(X) by Lemma A. 3.1. It therefore suffices to prove that fn —> f in C(X). But this follows easily from the proof of Lemma A. 3.1.
1.1. LINEAR SPACES
5
The spaces C(X] have the following interesting property that will be used quite frequently in our Chapter 4 on ANR-theory. Lemma 1.1.6. For every compact space X there are a compact space A and an imbedding i: X C(A) such that i[X] is linearly independent. Remark 1.1.7. The linear independence of the set i[X] in the above result is quite interesting, and has proved to be useful in several research papers in infinite-dimensional and related topology. Proof. Let Y be the topological sum of X and a point XQ £ X, and let Q be an admissible metric for Y. Let A be the subspace of C(Y) consisting of all Lipschitz functions /: Y -> R such that f ( x 0 ) = 0. Observe that if / G A then f[Y] C [-diam(y),diam(T)]. This is clear since if y G Y then
l/(i/)l = I/to) - o = l/(y) - /0*o)| < e(y,*o) < diam(r). It is easy to show that A is a closed subspace of C(Y). But even more is true. Claim 1. A is compact. Proof. Let (fm)m be any sequence in A. It suffices to prove that it has a convergent subsequence in A (Theorem A. 5.1). For every n let Un be a finite open cover of [— diam(F), diam(F)] consisting of open sets with diameter at most 7rn' . This cover has a Lebesgue number, say Xn > 0 (Lemma A. 5. 3). By compactness of X there is a finite open cover Vn of X such that mesh(V n ) < i/ 2 A n . Now fix any / G A and V G Vn. Since / is Lipschitz, the diameter of f[V] is at most Xn. Hence f[V] is contained in an element U G U n . So for each / G A there is a function £n(/): V n - > U n
such that for every V € Vn we have Let n = 1. Observe that there are finitely many functions Vi —> Ui only. Hence there is an infinite subset NI C N such that £i(/ m ) = £i(/fc) f°r all integers m, k 6 A7"! . Let m and k be two arbitrary elements of NI , and pick an arbitrary x 6 X. Pick an element F G Vi such that ar G I7. Then both fm(x) and /jfc(x) belong to the same element of Ui, i.e., |/m(^) ~ /fc( x )| < 2"1. Since x was arbitrary, this shows that ||/m — /^|| < 2"1. So now it is clear how to proceed. Let n\ = minA^i and consider the infinite set N\ \ {n\}. There is an infinite subset N% C NI\ {HI} such that for all ra and k in A^ the functions ^(/m) and £2 (/A:) agree. Then by a similar argument as the
6
1. BASIC TOPOLOGY
one above, ||/m — fk\\ < 2~ 2 for all m, k 6 N2. Let n2 = min 7V2. Continuing in this way resursively, we can construct an infinite sequence
ni < n2 < • • • < nm < • • • of natural numbers and a Cauchy sequence (fn )m in C(Y). Since C(Y) is a Banach space, this sequence has a limit and since A is closed, this limit belongs to A. So we conclude that (fnm)m is the desired convergent subsequence Of (fn}n0 Define i : X —> C(A) by the following formula:
t(*0(/) = /(*). Observe that i is well-defined. For i(x) should be an element of C(A), hence should be a function i(x) : A —> M. But an element of A is a function / from Y to R. So the formula tells us that i(x) sends the function / onto its evaluation in the point x £ X C Y. Claim 2. i: X —> i[X] is an isometry. Proof. Let £i,rr 2 G X. Then ||t(xi) - i(x2)\\ = sup{|t(*i)(/) - »
(1) Observe that the last inequality follows from the fact that / is Lipschitz. Define the function g : Y -> M by If y, y' 6 y are arbitrary then
\g(y) - g(y'}\ = \e(y,x2) = \Q(y,x2) -e(y',x2)\ < Q(y,y'}Also, g(xo) — 0. Hence g E A and -i(x2)(g}\ = \Q(xi,x2) - Q(x0, Hence by (1) we get ||Z(:EI) — z(^2)|| = ^(^15^2), as required.
0
It remains to prove that i[X] is a linearly independent subset of C(A). To this end, let #1, . . . , x n +i be distinct elements of ^C. We claim that z(x n +i) is not a linear combination of the i(xi), . . . , i(x n ). Define /i : Y -> E by = mm e ( y , X i ) . 0 [0, oo) is continuous, and aX is compact, it is clear that i[aX] is a bounded subset of C(A), hence so is i[X]. D If X is not compact then C(X) contains unbounded functions (see Exercise A.5.14 below), and so the formula \\f\\
=sup{\f(x)\:x£X}
does not define a norm on C(X). By considering the subset C*(X) of C(X) consisting of all bounded functions, this problem does not occur; C*(X) endowed with the sup-norm is a Banach space for the same reasons C(Y) is for compact Y. The topology defined here on C*(X) is called the topology of uniform convergence.
8
1. BASIC TOPOLOGY
Example 1.1.9. The space CQ. Put CQ = {x E s : linin-j.oo xn = 0}, and endow it with the norm \\x\\ = sup{|xn| : n E N}. It follows by straightforward calculations that this is indeed a norm compatible with the linear structure on CQ. There is however another way of proving this. Let S be a nontrivial convergent sequence including its limit t, and consider L = {/ E C(S) : /(*) - 0}. Then L is a closed linear subspace of C(S) which clearly can be identified with CQ. So c0 is a closed linear subspace of the function space C(S). The set CQ endowed with a different vector space topology will play a prominent role in our analysis of function spaces later. See Chapter 6 for details. Example 1.1.10. The Hilbert space I2. We saw that the topology on s is very different from the topology on any of its finite dimensional analogs W1. We shall now construct another natural 'limit' of the spaces W1 which behaves better (in this respect). Consider the usual inner product on En given by n (X \ 5I// y/ —
7/ j OC '11' iyi'
If we try to generalize this inner product for E°° then we have to deal with infinite series and it is therefore quite natural to restrict ourselves to the following subset of M°°: oo
I2 — {x E s : /]x2 < oo}. We shall first prove that t2 is a linear subspace of M°°. For every x E t2 we write p(x) = v/X^i X1- ^ x-> V e ^•> tnen Schwarz's inequality applied to En shows that | 5^=i ^^1 — P(X} ' P(y}- From this it follows that
P(x] -p(y] < oo, 1=1 and so oo
since all infinite series considered are convergent. We conclude that for every x, y G I2 we have x + y G i2 . If x e i2 and t G M then trivially tx € I2. Consequently, i2 is indeed a linear subspace of M°° .
1.1. LINEAR SPACES
Since X^i xiVi < °° f°r a^ X->V tion { - , - ) : i2 x £ 2 -^M,
e
^
we
have a well-defined func-
1=1 which is easily seen to be an inner product. Consequently, \\x\\ — p(x] defines a norm on I2 and the metric derived from this norm is: oo
£' We endow I2 with the topology generated by this metric and refer to I2 with this topology as Hilbert space. It is clear that a is a subset of I2 and so Lemma 1.1.4 shows that the topology that i2 inherits from s is different from the topology on i2 which we just defined. We will comment on the precise relation between these topologies later. Lemma 1.1.11. The metric g on I2 defined above is complete. So i2 is also a Banach space. But there is a difference with the spaces C(X] that we discussed before. The norm on i2 is derived from an inner product, while no inner product on C(I) yields its standard norm (Exercise 1.1.9). The topology on s is the topology of 'coordinatewise convergence', see Exercise A.2.2. Topologists usually find such product topologies easier to handle than topologies derived from a norm. However, convergence in £2 can be handled with the same ease, as is stated in the next result. Lemma 1.1.12 (Exercise 1.1.26). Suppose that (x(n)] and x G I2. The following statements are equivalent:
is a sequence in I2,
(1) linin-^oo x(n) = x (in I2), (2) limn_>.00 ||x(n)|| = ||x|| and for every i 6 N, lim^-^oo x(n}i = Xi. From Lemma 1.1.12 it immediately follows that the topology on I2 is finer than the topology that I2 inherits from s. However, more can be concluded. For example, consider the unit sphere 5 = { x e £ 2 : | x | | = l}. Since all points in 5 have the same norm, the topology that 5 inherits from f2 is precisely the same as the topology that S inherits from s, i.e., the topology of 'coordinatewise convergence'. This remark plays an important role in the proof of the Anderson Theorem from ANDERSON [15] that I2 and s are topologically homeomorphic (see also VAN MILL [298, Chapter 6] for a complete proof of this result).
10
1. BASIC TOPOLOGY
Classical theorems. We now present some classical theorems on Banach spaces that will be important later. Open Mapping Theorem 1.1.13. Let T be a continuous linear mapping of a Banach space E onto a Banach space F. Then T is open. Proof. The proof is in three steps. Claim 1. There exists a > 0 such that such that {y € F : \\y\\ < 1} C T[{x G £ : ||x|| < a}].
Proof. For each a > 0 put Ba = {x G -E1 : ||x|| < a}. Since oo
F
=U
n=l
and F is a Baire space (see Page 3), there exists m G N such that T[Bm] has nonempty interior. Since T is linear, it follows easily that T[Bm] is convex, and by the continuity of the algebraic operations on F, so is T[Bm}. In addition, Bm is symmetric, i.e., — Bm = Bm. Again since T is linear, it follows that T[Bm] is symmetric, from which it follows easily that T[Bm] is symmetric as well. Now choose y G F and /3 > 0 such that D(y,/3) C T[Bm]. Letz G F be such that \\z\
