The Mathematical Legacy

Transcription

1 The Mathematical Legacy - of Eduard Cech Edited by Miroslav Katetov Petr Simon 1993 Birkhauser Verlag Basel. Boston. Berlin

2 Editors Miroslav Katetov Matematicky ustav UK Sokolovski Praha 8 Czech Republic Petr Simon Matematicky ustav UK Sokolovska Praha 8 Czech Republic Reviewers Prof. RNDr. vera Trnkova, DrSc. Prof. RNDr. Oldnch Kowalski, DrSc. Co-edition by Birkhiiuser Verlag AG, Basel, Switzerland, and Academia, Publishing House of the Academy of Sciences of the Czech Republic, Prague, Czech Republic Exclusive distribution rights worldwide: Birkhiiuser Verlag AG, Basel, Switzerland with the exception of Albania, Bulgaria, China, Cuba, Czech Republic, Hungary, Mongolia, North Korea, Poland, Rumania, Slovak Republic, Vietnam, and countries of the former USSR and Yugoslavia, for which rights are held by Academia, Publishing House of the Academy of Sciences of the Czech Republic, Prague, Czech Republic Library of Congress Cataloging-in-Publication Data The Mathematical legacy of Eduard Cech/edited by Miroslav Katetov and Petr Simon. p. cm. Includes bibliographical references and index. 1. Algebraic topology. 2. Qeometry, Differential. 3. Dimension theory (Topology) 4. Stone-Cech compactifications. 5. Cech, Eduard, I. Katetov, Miroslav. II. Simon, Petr, QA612.M '.2-dc20 Deutsche Bihliothek Cataloging-in-Publication Data The mathematical legacy of Eduard Cech/ ed. by Miroslav Katetov and Petr Simon. - Basel; Boston; Berlin; Birkhiiuser, 1993 NE: Katetov, Miroslav [Hrsg.) This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of reprinting, re-use of illustrations, recitation, broadcasting, reproduction on microfilms or in other ways, and storage in data banks. For any kind of use permission of the copyright owner must be obtained. Miroslav Katetov, Petr Simon et ai., 1993 Softcover reprint of the hardcover 1st edition 1993 Translations Petr Simon, Jifi Vanzura, 1993 Camera-ready copy prepared by the authors in AMS-T EX ISBN ISBN (ebook) DOl / I

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4 EDUARD CECH

5 Foreword The work of Professor Eduard Cech had a si~ificant influence on the development of algebraic and general topology and differential geometry. This book, which appears on the occasion of the centenary of Cech's birth, contains some of his most important papers and traces the subsequent trends emerging from his ideas. The body of the book consists of four chapters devoted to algebraic topology, Cech-Stone compactification, dimension theory and differential geometry. Each of these includes a selection of Cech's papers, a brief summary of some results which followed from his work or constituted solutions to the problems he posed, and several selected papers by various authors concerning the areas of study he initiated. The book also contains a concise biography borrowed with minor changes from the book Topological papers of E. tech, a list of Cech's publications and a very brief note on his activity in the didactics of mathematics. The editors wish to express their sincere gratitude to all who contributed to the completion and publication of this book. The volume, with the exception of reprinted papers, has been typeset in AMS- TEX Miroslav Katetov and Petr Simon Prague, February 24, 1993

6 Contents Life and work of Eduard Cech. By M. Katetov, J. Novak and A. Svec Bibliography of E. Cech Cech-Stone Compactification. By P. Simon E. tech, On Bicompact Spaces, Annals of Mathematics 38, 1937 B. POSPiSIL, Remark on Bicompact Spaces, Annals of Mathematics 38, GELFAND AND A. KOLMOGOROFF, On rungs of Continuous Functions on Topological Spaces, Comptes Rendus (Doklady) de l'academie des Sciences de l'urss 22, GLICKSBERG, Stone-Cech Compactifications of Products, Transactions of Amer. Math. Soc. 90, W. Ru DIN, Homogeneity Problems in the Theory of Cech Compactifications, Duke Math. J. 23, PAROVICENKO, On a Universal Bicompactum of Weight N, Doklady Akad. Nauk. SSSR 150, (Translated from Russian by P. Simon) Z. FRoLlK, Non-Homogeneity of /3P - P, Comment. Math. Univ. Carolinae 8, K. KUNEN, Weak P-points in N, Colloquia Math. Soc. J. Bolyai 23, Dimension Theory. By M. Katetov 109 E. tech, On the Dimension of Perfectly Normal Spaces, Bull. Intern. Acad. Tcheque Sci. 33, (Translated from French by J. Vanzura) 130 E. tech, Contribution to Dimension Theory, Casopis Pest. Mat. Fys. 62, (Translated from Czech by P. Simon) O. V. LOKUCIEVSKIJ, On the Dimension of Bicompacta, Doklady Akad. Nauk SSSR 67,1949. (Translat/id from Russian by P. Simon) 161 C. H. DOWKER, Inductive Dimension of Completely Normal Spaces, Quart. J. Math. Oxford Ser. (2) 4, C. H. DOWKER AND W. HUREWICZ, Dimension of Metric Spaces, Fundamenta Mathematicae 43,

7 P. VOPENKA, On the Dimension of Compact Spaces, Czechoslovak Math. J. 8, (Translated from Russian by P. Simon) V. V. FILIPPOV, Bicompacta with Distinct Dimensions ind and dim, Doklady Akad. Nauk. SSSR 192, (Translated from Russian by P. Simon) E. POL AND R. POL, A Hereditarily Normal Strongly Zero-Dimensional Space with a Subspace of Positive Dimension and an N-Compact Space of Positive Dimension, Fundamenta Mathematicae 97, M. G. CHARALAMBOUS, Spaces with Noncoinciding Dimensions, Proceedings of Amer. Math. Soc. 94, Algebraic Topology. By E. G. Sklyarenko 213 E. CECH, General Homology Theory in an Arbitrary Spaie, Fundamenta Mathematicae 10, (Translated from French by J. Vanzum) 231 E. CECH, Betti Groups of an Infinite Complex, Fundamenta Mathematicae 25, (Translated from French by J. Vanium) E. CECH, Multiplications On a Complex, Annals of Math. 37, S. LEFSCHETZ, On Generalized Manifolds, American J. of Math. 55, C. H. DOWKER, Cech Cohomology Theory and the Axioms, Annals of Math. 51, Differential Geometry. By I. Kolar 333 E. CECH, On the Surfaces All Segre Curves of Which Are Plane Curves, Publ. Fac. Sci. Univ. Masaryk 11, (Tmnslated from French by J. Vanium) E. CECH, Developable Transformations of Line Congruences, Czechoslovak Math. J. 6, (Translated from French by J. Vanzum) A. SVEC, On the Differential Geometry of a Surface Embedded in a Three Dimensional Space With Projective Connection, Czechoslovak Math. J. 11, (Tmnslated from French by J. Vanium) I. Ko LA.ii, Order of Holonomy of a Surface With Projective Connection, Casopis Pest. Mat. 96, B. CENKL, Geometric Deformations of the Evolution Equations and Backlund Transformations, Physica 18D, Professor Cech and Didactics of Mathematics. By E. Kmemer Subject Index Acknowledgement