Table Of ContentTOPOLOGY
Other books in this series are
The Theory of Groups: An Introduction
Joseph J. Rotman
A Survey of Matrix Theory and Matrix Inequalities
Marvin Marcus and Henryk Minc
THis Book Is PArRT OF THE
ALLYN AND BACON SERIES IN ADVANCED MATHEMATICS
CoNSULTING EDITOR: IRVING KAPLANSKY
UNIVERSITY OF CHICAGO
Other books in this series are
The Theory of Groups : An Introduction
Joseph J. Rotman
A Survey of Matrix Theory and Matrix Inequalities
Marvin Marcus and Henryk Minc
TOPOLOGY
James Dugundji
Professor of Mathematics
University of Southern California
Los Angeles
ALLYN AND BACON, INC.
BOSTON « LONDON . SYDNEY « TORONTO
To Merope
© Copyright 1966 by Allyn and Bacon, Inc.
470 Atlantic Avenue, Boston.
All rights reserved.
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 informational storage
and retrieval system, without written
permission from the copyright owner.
Library of Congress Catalog Card Number: 66-10940
Printed in the United States of America
ISBN " 0-205-00271-4
Twelfth printing . . . June, 1978
Preface
viii
concepts discussed in the book. Two appendices, one on linear topo-
logical spaces and the other on limit spaces, are included.
Nearly every definition is followed by examples illustrating the use of
the abstract concept in some fairly concrete situations. This device
makes the book suitable for self-study. It also enables the instructor who
uses the book as a text to proceed rapidly to those parts of the subject
that he deems of greater importance.
Remarks, in small type, call attention either to further developments,
or to direct applications in other branches of mathematics.
The problems, which are given at the end of each chapter, can all be
solved by the methods developed in the book. Moreover, no proof in
the text relies on the solution of any problem. Some of the problems are
routine. Others are important theorems that complement the material
in the text; these are accompanied by hints for their solution.
The notation of symbolic logic used throughout the book is given
immediately after the table of contents.
I wish to thank E. A. Michael and Ky Fan, who read the original
manuscript, for their valuable suggestions; and H.-J. Groh and P. A.
White, for their help with the proofreading. I am particularly indebted
to H. Salzmann for his constant willingness to discuss points of detail
and content: his imaginative and penetrating criticisms and suggestions
have led to many improvements.
I also wish to thank Mrs. L. Syfritt, for her tidy and meticulous typing
work; and the members of Allyn and Bacon who were involved with this
book, for their patience and co6peration. Finally, I gratefully acknowl-
- edge the support given me by the National Science Foundation during
the period that this book was being written.
James Dugundji
The University of Southern California
Contents
I. Elementary Set Theory l
1 Sets
1
2 Boolean Algebra
3
3 Cartesian Product
7
4 Families of Sets
8
5 Power Set
10
6 Functions, or Maps ~ 10
7 Binary Relations; Equivalence Relations 14
8 Axiomatics
17
9 General Cartesian Products 21
Problems
25
ll. Ordinals and Cardinals » 29
1 Orderings
29
2 Zorn’s Lemma; Zermelo’s Theorem
31
3 Ordinals
36
4 Comparability of Ordinals ' 38
5 'Transfinite Induction and Construction 40
6 Ordinal Numbers
41
7 Cardinals
45
8 Cardinal Arithmetic
49
9 'The Ordinal Number £
54
Problems
57
Contents
X
62
lll. Topological Spaces
62
1 Topological Spaces
64
2 Basis for a Given Topology
65
3 Topologizing of Sets
68
4 Elementary Concepts
5 'Topologizing with Preasmgned Elementary Opera-
72
tions
74
6 G, F, and Borel Sets
77
7 Relativization
78
8 Continuous Maps '
81
9 Piecewise Definition of Maps ‘
83
10 Continuous Maps into E?
86
11 Open Maps and Closed Maps
87
12 Homeomorphism
90
Problems
98
IV. Cartesian Products
98
1 Cartesian Product Topology
101
2 Continuity of Maps
103
3 Slices in Cartesian Products
104
4 Peano Curves
105
Problems
107
V. Connectedness
107
1 Connectedness
110
2 Applications
111
3 Components
113
4 Local Connectedness-
114
5 Path-Connectedness
116
Problems
VI. Identification Topology; Weak Topology 120
120
1 Identification Topology
122
2 Subspaces
123
3 General Theorems
125
4 Spaces with Equivalence Relations
126
5 Cones and Suspensions
