Topology PDF
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TOPOLOGY 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
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