Dimension Theory RYSZARD ENGELKING Warsaw University 1978 NORTH-HOLLAND PUBLISHING COMPANY AMSTERDAM . OXFORD. NEW YORK PWN - POLISH SCIENTIFIC PUBLISHERS WARSZAWA A revised and enlarged translation of Teoria wymiaru, Warszawa 1977 Translated by the Author Library of Congress Cataloging in Publication Data Ryszard, Engelking. Dimension theory. Translation of Teoria wymiaru. Bibliography: p . 1. Topological spaces. 2. Dimension theory (Topology) I. Title. QA611.3.R9713 514.3 ISBN 0-444-85176-3 Copyright 78-12442 @ by PWN - Polish Scientific Publishers - Warszawa 1978 All rights reserved. No part of this publication may be reproduced, stored in a retrieval system or transmitted in any form or by any means, electronic, mechanical, photocopying, recording, or otherwise, without the prior written permission of the copyright owner Published in co-edition with PWN-Polish Scientific Publishers - Warszawa 1978 Distributors for the socialist countries: ARS POLONA, Krakowskie Przedmiekie 7, 00-068 Warszawa, Poland North-Holland Publishing Company - Amsterdam Distributors outside the socialist countries: Elsevier/North-Holland Inc. . New 52, Vanderbilt Avenue, New York, N. Y. 10017, U.S.A. Sole distributors for the U.S.A. and Canada Printed in Poland by D.R.P. York . Oxford PREFACE Dimension theory is a branch of topology devoted to the definition and study of the notion of dimension in certain classes of topological spaces. It originated in the early twenties and rapidly developed during the next fifteen years. The investigations of that period were concentrated almost exclusively on separable metric spaces ; they are brilliantly recapitulated in Hurewicz and Wallman’s book Dimension Theory, published in 1941. After the initial impetus, dimension theory was at a standstill for ten years or more. A fresh start was made at the beginning of the fifties, when it was discovered that many results obtained for separable metric spaces can be extended to larger classes of spaces, provided that the dimension is properly defined. The
[email protected] reservation necessitates an explanation. It is possible to define the dimension of a topological space X in three different ways, the small inductive dimension indX, the large inductive dimension IndX, and the covering dimension dimX. The three dimension functions coincide in the class of separable metric spaces, i.e., indX = IndX = dimX for every separable metric space X. In larger classes of spaces the dimensions ind, Ind, and dim diverge. At first, the small inductive dimension ind was chiefly used; this notion has a great intuitive appeal and leads quickly and economically to an elegant theory. The dimension functions Ind and dim played an auxiliary role and often were not even explicitly defined. To attain the next stage of development of dimension theory, namely its extension to larger classes of spaces, first and foremost to the class of metrizable spaces, i t was necessary to realize that in fact there are three theories of dimension and to decide which is the proper one. The @option of such a point of view immediately led to the understanding that the dimension ind is practically of no importance outside the class of separable metric spaces and that the dimension dim prevails over the dimension Ind. The greatest achievement in dimension theory during the fifties was the discovery that IndX = dimX for every metric space X and the creation of a satisfactory dimension theory for metrizable spaces. Since that time many important results on dimension of topological spaces VI Preface have been obtained; they primarily bear upon the covering dimension