Table Of ContentLecture Notes in Mathematics 1652
Editors:
A.Dold, Heidelberg
F.Takens,Groningen
Springer
Berlin
Heidelberg
New York
Barcelona
Budapest
HongKong
London
Milan
Paris
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Singapore
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Stevo Todorcevic
Topics in Topology
Springer
Author
StevoTodorcevic
DepartmentofMathematics
UniversityofToronto
Toronto,CanadaM5S 3G3
e-mail: [email protected]
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Mathematics SubjectClassification(1991):
54-02, 54C35, 54Hll, 22D05, 26A21, 28A60, 54B20, 54E50
ISSN0075-8434
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PREFACE
Sixty or seventy years ago Topology was a very well defined area of mathe
matics but after the appearance of great many branches of this field the name
started to have as many different interpretations except perhaps at the very
elementary level. For example, a recent Proceedings of a Topology Conference
contains papers from homotopy theory, continua theory, infinite dimensional
topology, metrization theory, topological algebra, and more. Thus, choosing a
material for a relatively short graduate course entitled "Advanced Topology"
was quite a challenge. The following is a reworked version of a set of lecture
notes from that course taken by a group of excellent and highly motivated stu
dents of the University of Toronto: Ilijas Farah, James Hirschorn, David Kerr,
Andrzej Pienkowski, and Shaloub Razak. We are also grateful to Max Burke,
Claude Laflamme and Jianping Zhu for reading parts of this material and to
Boban Velickovic for suggesting Exercise 3in the section on Ptak's Lemma. The
greatest debt weowe to Ilijas Farah for a considerable technical (and mathemat
ical) help in preparing the final version of the manuscript. This, of course, are
not the people to be blamedfor shortcomingsof this book; the responsibilityfor
this is solely mine. Here is a short overview of the content of this book which is
naturally divided into four independent chapters.
Part Itouches the FunctionSpaceTheory, the placewhere theTopologymeets
Analysis. We give a proof of the BourgainFremlinTalagrand theorem about
compact sets of Baire Classl functions, Namioka's Theorem about the joint
and separate continuity in products, and Ptak's approach to study of weakly
compact sets in Banach spaces.
In Part II westudy a semigroupstructure on the CechStone compactification
of the integers in the course ofpresenting somerecent applicationsof topological
methods to Combinatorics and Number Theory, such as the work of S. Glazer,
N. Hindman, H. Furstenberg and others.
Part III deals with compact and compactly generated topological groups.
The central result is the classical IvanovskiiVilenkinKuzminov Theorem that
compact groups are dyadic, but we also present some recent results such as the
countable chain condition theorem of Tkachenko.
In Part IV westudy hyperspaces ofcompact metricspaces. We have managed
to present Marjanovic's classification of hyperspaces of zerodimensional metric
compacta. We also give a description of the multiplication table for this class of
metric compacta.
There is a number of exercises scattered throughout the text. They serve
several purposes. For example, an exercise mightbe quite easy and it could help
VI
the reader to recheck his or her understanding ofthe text that comesjust before.
There are exercises which might look quite challenging and their purpose is to
inform the reader about the material which we chose not to present here. The
details can usually be found in the references listed at the end of each Chapter.
S. Todorcevic
Toronto, Dec. 1995
CONTENTS
I COMPACT SETS IN FUNCTION SPACES 1
1 Topology of pointwise convergence 3
2 A theorem of Eberlein 8
3 Ptak's Lemma 12
4 Namioka's theorem ,.17
5 Rosenthal's dichotomy 20
6 Properties of Baire and Ramsey 25
7 Baire property of analytic sets 29
8 Baire property offilters and ideals 32
9 Selective coideals 35
10 Baire's characterization theorem and its corollaries 41
11 Borel sets 46
12 A selective analytic ideal 49
13 Bourgain-Fremlin-Talagrand's theorem 55
References for Chapter I 59
II THE SEMIGROUP ,BJ'::I 61
14 A space of ultrafilters 63
15 Glazer's Theorem 67
16 A topological proofof the van der Waerden theorem 71
17 A semigroup of variable words 75
References for Chapter II 78
III COMPACT AND COMPACTLY GENERATED GROUPS 79
18 Countable chain conditions of topological groups 81
19 Michael's selection theorem 87
20 Inverse systems 91
21 Haydon's theorem 95
22 Quotient groups 99
23 A decomposition of compact groups 105
24 Pestov's theorems 109
25 Free topological groups 113
References for Chapter III 119
IV HYPERSPACES 121
26 Exponentially complete spaces 123
27 Vaught's homeomorphismtheorem 127
28 Resolving a space: Accumulation orders and spectra 130
29 Accumulation spectra of hyperspaces , 135
30 List of all exponentials , 139
VIII
31 Multiplication of accumulation orders 141
References for Chapter IV 146
Index 149
Index of special symbols 152
CHAPTER I
COMPACT SETS IN FUNCTION SPACES
Weakly compact sets in Banach spaces have received a considerable attention
also by topologists ever since Amir and Lindenstrauss proved in 1968 their fa-
mous characterization of these sets as compacta which can be embedded into
Tychonoff cubes in a very special way. This replacement of the weak topology
by the Tychonoff topology (or, in other words, by the topology of pointwise
convergence) has resulted into a number of internal characterizations of these
spaces as well as into a discovery of several related classes of compacta. In this
Chapter we shall consider a single topological property of some of these classes
of compacta, the property of sequential compactness. This property has become
of a considerable interest to both analysts and topologists especially after Odell
and Rosenthal have interpreted the famous Rosenthal tl theorem as saying that
a separable Banach space E contains no copy of tl if and only if the sequential
closure of E (viewed as a set of continuous functions on the dual ball of E with
the weak* topology) is equal to its second dual E**, and after Bourgain, Fremlin
and Talagrand proved that sequential convergence determines also the topology
of every compact set of Baire class 1 functions. We shall prove some of these
results here. Our approach in finding a sequence which converges to a point
in the closure of a set might be novel as it involves a combinatorial theorem of
A.R.D. Mathias about selective coideals on the set of integers. Given a sequence
{f, } which accumulates to f, one is naturally led to the coideal of all sets M of
2 Chapter I
MEn}nf{
integers for which the corresponding subsequence also accumulates to
f. What we need is an infinite set of integers all of whose infinite subsets belong
to the eoideal. But this kind of conclusion is the subject matter of a rather large
family of combinatorial theorems to which Mathias' result belongs.
We shall also treat Pt$k's approach to uniform convergence of convex com-
binations of elements of a fixed sequence of continuous functions which point-
wise accumulates to some continuous function. This will be our way to proving
Namioka's theorem about separate versus joint continuity in product spaces.
The theorems of Eberlein and Grothendieck about (countable) compactness
)X(pC
in are really results whose proper context is the theory of topological
vector spaces. We have included them here as they serve as a historically very
early hints to the more general and powerful theory of convergence in function
spaces that we know today.
