Table Of ContentSpringer Undergraduate Mathematics Series
Springer-Verlag London Ltd.
Advisory Board
Professor P.J. Cameron Queen Mary and Westfield College
Dr M.A.J. Chaplain University of Dundee
Dr K. Erdmann Oxford University
Professor L.C.G. Rogers University of Bath
Dr E. Silli Oxford University
Professor J.F. Toland University of Bath
Other books in this series
T.S. Blyth and E.F. Robertson: Basic Linear Algebra (3-540-76122-5)
Z. Brzemiak and T. Zastawniak: Basic Stochastic Processes (3-540-76175-6)
P.J. Cameron: Sets, Logic and Categories (1-85233-056-2)
M. Capi.Dksi and E. Kopp: Measure, Integral and Probability (3-540-76260-4)
S. Dineen: Multivariate Calculus and Geometry (3-540-76176-4)
D.L. Johnson: Elements of Logic via Numbers and Sets (3-540-76123-3)
G.A. Jones and J.M. Jones: Elementary Number Theory (3-540-76197-7)
G.S. Marshall: Introductory Mathematics: Applications and Methods (3-540-76179-9)
P.C. Matthews: Vector Calculus (3-540-76180-2)
G. Smith: Introductory Mathematics: Algebra and Analysis (3-540-76178-0)
D.A.R. Wallace: Groups, Rings and Fields (3-540-76177-2)
loan James
Topologies and
Uniformities
With 25 Figures
Springer
loan Mackenzie James, MA, DPhil, FRS
Mathematical Institute, Oxford University, 24-29 St Giles, Oxford OXl 3LB, UK
Cover illustration elements reproduced by kind permission of:
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American Statistical Association: Chance Vol 8 No I, 1995 article by KS and KW Heiner 'Tree Rings of the Northern Shawangunks' page 32 fig 2
Springer-Verlag: Mathematica in Education and Research Vol4 Issue 3 1995 article by Roman E Maeder, Beatrice Amrhein and Oliver Gloor
'Illustrated Mathematic~ Visualization of Mathematical Objects' page 9 fig II, originally published as a CD ROM 'lllustrated Mathematics'
by TEWS: ISBN 0-387-14222-3, german edition by Birkhauser: ISBN 3-7643-5100-4.
Mathematics in Education and Research Vol 4 Issue 3 1995 article by Richard J Gaylord and Kazume Nishidate 'Traffic Engineering with
Cellular Automata' page 35 fig 2. Mathematica in Education and Research Vol 5 Issue 2 1996 article by Michael Trott 'The lmplicitization
of a Trefoil Knot' page 14.
Mathematica in Education and Research Vol 5 Issue 2 1996 article by Lee de Cola 'Coins, Trees, Bars and Bells: Simulation of the Binomial
Process page 19 fig 3. Mathematica in Education and Research Vol 5 Issue 2 1996 article by Richard Gaylord and Kazume Nashidate
'Contagious Spreading' page 33 fig I. Mathematica in Education and Research Vol 5 Issue 2 1996 article by Joe Buhler and Stan Wagon
'Secrets of the Madelung Constant' page 50 fig I.
ISBN 978-1-85233-061-3
British Library Cataloguing in Publication Data
James, loan M.
Topologies and uniformities
I. Topology 2. Topological spaces
I. Title
514
ISBN 978-1-85233-061-3 ISBN 978-1-4471-3994-2 (eBook)
DOi 10.1007/978-1-4471-3994-2
Library of Congress Cataloging-in-Publication Data
James, I. M. (loan Mackenzie), 1928-
Topologies and uniformities/I. M. James.
p. em.
Includes bibliographical refrerences and index.
ISBN 978-1-85233-061-3
I. Topology. 2. Topological spaces. 3. Uniform spaces.
I. Title.
QA611.J337 1999 98-41748
514-dc21 CIP
Apart from any fair dealing for the purposes of research or private study, or criticism or review, as permitted
under the Copyright, Designs and Patents Act 1988, this publication may only be reproduced, stored or
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Licensing Agency. Enquiries concerning reproduction outside those terms should be sent to the publishers.
©Springer-Verlag London 1999
Originally published by Springer-Verlag London Berlin Heidelberg in 1999
Expanded and revised version of Topological and U11ijorm Spaces, published by
Springer-Verlag London Ltd.
The use of registered names, trademarks etc. in this publication does not imply, even in the absence of a
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Contents
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . .. . . . . . . . . . . . .. . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix
Historical Note .............................................................................................. xiii
Chapter 1 Topological Spaces..................................................................... 1
1.1 General remarks . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . .. . . . . . . . . . . .. . . . . . . . . 1
1.2 Direct and inverse images.......................................................... 2
1.3 Cartesian products.................................................................... 3
1.4 Relations.................................................................................... 4
1.5 Axiotns of topology.................................................................... 5
1.6 Closure and interior................................................................... 8
1. 7 Generating families.................................................................... 11
1.8 Ordered sets............................................................................... 12
1.9 Filters ........................................................................................ 14
1.10 Neighbourhoods ......................................................................... 17
1.11 Metric spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . .. . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . 19
1.12 Filters on topological spaces...................................................... 23
Exercises . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . .. . . . . . . . . . . . . . . 26
Chapter 2 Continuity . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . .. . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
2.1 General remarks . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . .. . . . . . . . . . . . .. . . . . .. . . . . . . . . .. . . . . 29
2.2 Continuous functions................................................................. 30
2.3 Homeomorphisms . .. . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
2.4 The topological product . . . . . . . . . . . . . . . . . . . . .. . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . .. .. 35
2.5 Topological groups . . . . . .. . . . . .. . . . .. . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . .. . . . . . . . . . . . .. . 42
Exercises . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . .. . . . . . . . . . . . .. . . . . . . .. .. . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . 45
v
Topologies and Uniformities
VI
Chapter 3 The Induced Topology and Its Dual . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 7
3.1 The induced topology................................................................ 4 7
3.2 Subspaces.... .. .. . . . . .. . . . . . .. . . . . . ... .. . . . . . . . . . . . . . . . .. . . . . . .. . . . . .. . . . .. . . . . . . . . . . . . . . . . . 51
3.3 The co induced topology .. . . . .. .. . .. .. . . . .. . . . . . . . . . . . . . .. . . . . . . . . . .. . . . . . . . . . . . . . . . 56
3.4 Quotient spaces . . .. . . . . . . . .. . . . .. . . . .. .. . . . .. . . . . .. .. . . . . . . . . . . . .. . . . .. . . . . . .. . . . . . . . . . . 58
Exercises............................................................................................. 60
Chapter 4 Open Functions and Closed Functions ...................................... 63
4.1 General remarks . . . . . .. . . . . . . . .. .. .. . . . .. .. . . . . . . . .. . . . .. . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . 63
4.2 Open functions . . . . . . . . . . . . . . . . .. .. . . . . . .. . . . . . . . . . .. . . . .. . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . 63
4.3 Closed functions . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . 6 7
Exercises . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . .. . . . . . . .. . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . 72
Chapter 5 Compact Spaces ......................................................................... 75
5.1 Introduction............................................................................... 75
5.2 Compactness via filters.............................................................. 79
5.3 Compactness via coverings........................................................ 83
Exercises............................................................................................. 85
Chapter 6 Separation Conditions............................................................... 87
6.1 General remarks........................................................................ 87
6.2 T spaces.................................................................................... 87
1
6.3 Hausdorff spaces........................................................................ 88
6.4 Regular spaces .. . . . . .. .. . . . .. . . . . . .. .. ... .. . . . .. . . . . . . . . .. . . . .. . . . . .. . . . . . . . . . . . . . . . . . . . . 94
6.5 Normal spaces............................................................................ 97
6.6 Compactly regular spaces .......................................................... 100
Exercises ............................................................................................. 102
Chapter 7 Uniform Spaces .......................................................................... 103
7.1 Uniform structures .................................................................... 103
7.2 Separated uniformities .............................................................. 108
7.3 Totally bounded uniformities .................................................... 108
7.4 Uniform continuity .................................................................... 110
7.5 Uniform equivalences ................................................................. 112
7.6 The uniform product ................................................................. 113
7.7 The induced uniformity ............................................................. 115
7.8 The coinduced uniformity? ........................................................ 120
Exercises ............................................................................................. 124
Chapter 8 The Uniform Topology .............................................................. 127
8.1 Uniform neighbourhoods ........................................................... 127
Contents vii
8.2 Closure in uniform spaces .......................................................... 131
8.3 Uniformization of compact Hausdorff spaces ............................ 134
8.4 Cauchy sequences ...................................................................... 136
8.5 Cauchy filters ............................................................................ 138
8.6 Uniformization of function spaces ............................................. 141
Exercises ............................................................................................. 144
Chapter 9 Connectedness ........................................................................... 145
9.1 Connected spaces ....................................................................... 145
9.2 Connectedness components ....................................................... 150
9.3 Locally connected spaces ........................................................... 152
9.4 Pathwisc-connected spaces ........................................................ 152
9.5 Uniformly connected spaces ...................................................... 159
9.6 Uniformly locally connected spaces ........................................... 161
Exercises ............................................................................................. 163
Chapter 10 Countability and Related Topics .............................................. 165
10.1 Countability .............................................................................. 165
10.2 Lindelof spaces ........................................................................... 166
10.3 Countably compact spaces ........................................................ 168
10.4 Sequentially compact spaces ...................................................... 170
10.5 Separable spaces ........................................................................ 173
Exercises ............................................................................................. 176
Chapter 11 Functional Separation Conditions ............................................. 1 79
11.1 General remarks ........................................................................ 179
11.2 Completely regular spaces ......................................................... 179
11.3 Uniformizability ......................................................................... 181
11.4 The Urysohn theorem ................................................................ 183
11.5 The Tietze theorem ................................................................... 185
Exercises ............................................................................................. 187
Chapter 12 Completeness and Completion .................................................. 189
12.1 Complete uniform spaces ........................................................... 189
12.2 Metric completion ...................................................................... 197
12.3 Uniform completion ................................................................... 199
Exercises ............................................................................................. 200
Select Bibliography ....................................................................................... 203
Solutions to Exercises .................................................................................... 205
Index ............................................................................................................. 227
Introduction
This book is based on lectures I have given to senior undergraduate and graduate
audiences at Oxford and elsewhere over the years. My aim has been to provide an
outline of both the topological theory and the uniform theory, with an emphasis
on the relation between the two. Although I hope that the prospective specialist
may find it useful as an introduction it is the non-specialist I have had more
in mind in selecting the contents. Thus I have tended to avoid the ingenious
examples and counterexamples which often occupy much of the space in books
on general topology, and I have tried to keep the number of definitions down
to the essential minimum. There are no particular prerequisites but I have
worked on the assumption that a potential reader will already have had some
experience of working with sets and functions and will also be familiar with
the basic concepts of algebra and analysis.
An earlier version of the present book appeared in 1987 under the title Topo
logical and Uniform Spaces. When the time came for a new edition I came to
the conclusion that, rather than just making the necessary corrections, it
would be better to make more substantial alterations. Parts of the text have
been rewritten and new material, including new diagrams, added. The sets of
exercises at the end of each chapter have been revised and worked solutions to
all of them provided at the end of the book. Also a historical note has been
added and the Select Bibliography has been expanded. Altogether these changes
seemed sufficient to justify a new title.
The book divides naturally into three sections. Thus the first six chapters are
devoted to the topological theory while the next two are devoted to the uniform
ix
X Topologies and Uniformities
theory. The last four, which are independent of each other, draw on ideas from
both the topological section and the uniform section.
After a few preliminaries, in which notation and terminology are established,
the first chapter of the topological section mainly deals with the basic axioms.
Illustrations are taken from interval topologies and metric topologies, with
special reference to the real line. No previous knowledge of metric spaces is
assumed. An outline of the theory of filters is included.
The second chapter is concerned with continuity: topology is about con
tinuous functions just as much as topological spaces. The topological product
is dealt with here. Also topological groups are introduced at this stage both
because of their intrinsic interest and because they provide such excellent
illustrations of points in the general theory. Subspaces and quotient spaces are
considered in Chapter 3, with a wide range of examples.
Most accounts of the theory go on to discuss separation axioms, connectedness
and so forth at this point. But in my view compactness should come first, because
of its fundamental importance. I believe the concept arises most naturally from a
discussion of open functions and closed functions. This is not the orthodox
approach, of course, but I have tried to justify it by showing that all the usual
properties of compact spaces such as the Reine-Borel theorem can be proved
quite simply and directly from this approach. I also show how compactness
can be characterized in terms of filters, and incidentally show how the best
known characterization of compactness, in terms of open coverings, can be
obtained. The general Tychonoff theorem is proved, followed by some observa
tions on the subject of function spaces. This material occupies Chapters 4 and 5.
Chapter 6 is devoted to the separation axioms: the basic properties of
Hausdorff, regular and normal spaces are established. In a later chapter there
is an account of the corresponding functional separation axioms.
Chapter 7 of the uniform section deals with the basic axioms of uniform struc
ture, with illustrations from topological groups and metric spaces. I have tried to
show how the idea of a uniformity is a very natural one, in many ways more
natural than the idea of a topology. This leads on to the notion of uniform
continuity: the uniform theory is about uniformly continuous functions just as
much as uniform spaces. I also deal with the uniform product with subspaces
and, to a limited extent, with quotient spaces.
In Chapter 8 the connection between the uniform and the topological theories
is established. It becomes clear at this stage that results about topological groups
and metric spaces found earlier can be regarded as special cases of results about
uniform spaces. The chapter continues with a discussion of the Cauchy condi
tion, both for sequences and for filters. This lays the foundation for a subsequent
chapter on completeness and completion.
