The Mathematical Legacy of Eduard Čech

444 Pages·1993·20.19 MB·English

by  J. Novák, A. à vec (auth.), Petr Simon (eds.)

The Mathematical Legacy
-
of Eduard Cech
Edited by
Miroslav Katetov
Petr Simon
1993 Birkhauser Verlag
Basel . Boston . Berlin
Editors
Miroslav Katetov Petr Simon
Matematicky ustav UK Matematicky ustav UK
Sokolovski 83 Sokolovska 83
186 00 Praha 8 18600 Praha 8
Czech Republic 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, 1893-1960. I. Katetov, Miroslav. II. Simon, Petr, 1944-
QA612.M378 1993
514'.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 978-3-0348-7526-4 ISBN 978-3-0348-7524-0 (eBook)
DOl 10.1007/978-3-0348-7524-0
9 8 7 6 543 2 I
EDUARD CECH 1893-1960
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
Contents
Life and work of Eduard Cech. By M. Katetov, J. Novak and A. Svec 9
Bibliography of E. Cech 21
Cech-Stone Compactification. By P. Simon 26
E. tECH, On Bicompact Spaces, Annals of Mathematics 38, 1937 38
B. POSPiSIL, Remark on Bicompact Spaces, Annals of Mathematics 38, 1937 60
1. GELFAND AND A. KOLMOGOROFF, On rungs of Continuous Functions on
Topological Spaces, Comptes Rendus (Doklady) de l'Academie des
Sciences de l'URSS 22, 1939 ... ..... ..... .... ... ....... ..... ....... 62
1. GLICKSBERG, Stone-Cech Compactifications of Products, Transactions of
Amer. Math. Soc. 90, 1959 .......... ..... ..... ..... ....... .. . ..... 67
W. Ru DIN, Homogeneity Problems in the Theory of Cech Compactifications,
Duke Math. J. 23, 1956 ............................................ 81
1. 1. PAROVICENKO, On a Universal Bicompactum of Weight N, Doklady
Akad. Nauk. SSSR 150, 1963. (Translated from Russian by P. Si-
mon) .................... .......................................... 93
Z. FRoLlK, Non-Homogeneity of /3P - P, Comment. Math. Univ. Carolinae
8, 1967 .............. ........... ..... ..... ... .. ... .... ... .... . ..... 97
K. KUNEN, Weak P-points in N°, Colloquia Math. Soc. J. Bolyai 23,1978 100
Dimension Theory. By M. Katetov 109
E. tEcH, On the Dimension of Perfectly Normal Spaces, Bull. Intern. Acad.
Tcheque Sci. 33, 1932. (Translated from French by J. Vanzura) 130
E. tECH, Contribution to Dimension Theory, Casopis Pest. Mat. Fys. 62,
1933. (Translated from Czech by P. Simon) ........................ 149
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, 1953 . ..... ..... ... .. ..... .... ..... .. ... 165
C. H. DOWKER AND W. HUREWICZ, Dimension of Metric Spaces, Funda-
menta Mathematicae 43, 1956 . ..... ..... ... . . . . ... ..... .... .. .... . 178
P. VOPENKA, On the Dimension of Compact Spaces, Czechoslovak Math. J.
8, 1958. (Translated from Russian by P. Simon) . . . . . . . . . . . . . . . . . . . 184
V. V. FILIPPOV, Bicompacta with Distinct Dimensions ind and dim, Dok-
lady Akad. Nauk. SSSR 192, 1970. (Translated from Russian by P.
Simon) ............................................................ 191
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, 1977 196
M. G. CHARALAMBOUS, Spaces with Noncoinciding Dimensions, Proceedings
of Amer. Math. Soc. 94, 1985 ..................................... 204
Algebraic Topology. By E. G. Sklyarenko 213
E. CECH, General Homology Theory in an Arbitrary Spaie, Fundamenta
Mathematicae 10, 1932. (Translated from French by J. Vanzum) 231
E. CECH, Betti Groups of an Infinite Complex, Fundamenta Mathematicae
25, 1935. (Translated from French by J. Vanium) . . . . . . . . . . . . . . . . . . 256
E. CECH, Multiplications On a Complex, Annals of Math. 37, 1936 265
S. LEFSCHETZ, On Generalized Manifolds, American J. of Math. 55, 1933 282
C. H. DOWKER, Cech Cohomology Theory and the Axioms, Annals of Math.
51, 1950 ........................................................... 318
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, 1922. (Tmnslated from French by J.
Vanium) ........... ..... .......... ..... ..... ..... ..... ..... ....... 357
E. CECH, Developable Transformations of Line Congruences, Czechoslovak
Math. J. 6, 1956. (Translated from French by J. Vanzum) .......... 393
A. SVEC, On the Differential Geometry of a Surface Embedded in a Three
Dimensional Space With Projective Connection, Czechoslovak Math.
J. 11, 1961. (Tmnslated from French by J. Vanium) ............... 416
I. Ko LA.ii , Order of Holonomy of a Surface With Projective Connection,
Casopis Pest. Mat. 96, 1971 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 428
B. CENKL, Geometric Deformations of the Evolution Equations and Backlund
Transformations, Physica 18D, 1986 .... ..... ..... ..... ..... ....... 436
Professor Cech and Didactics of Mathematics. By E. Kmemer 439
Subject Index 442
Acknowledgement 444
Life and Work of Eduard Cech
Eduard tech, professor of Charles University, and member of the Czechoslovak Aca-
demy of Sciences, was the greatest Czechoslovak mathematician and one of the leading
world specialists in the fields of differential geometry and topology. To these fields he
contributed works of basic importance.
He was born on June 29, 1893, in Stracov in northeastern Bohemia. He attended
the secondary school in Hradec Knllove. In 1912 he began to study mathematics at
Charles University in Prague. He learnt most of his mathematics in the library of the
Union of Czech Mathematicians and Physicists. Within the period of five semesters
he studied thoroughly a considerable amount of mathematical literature of his own
choice aJ;ld acquired knowledge in a number of mathematical disciplines without any
guidance. While studying some treatises on elementary mathematics he discovered
logical gaps in the proofs; he took a special liking for correcting and completing them.
This was the origin of his interest in didactic questions of mathematics. As at that
time two fields of study were required for the position of the secondary school teacher,
he chose as the other subject descriptive geometry and devoted himself to the study
of different branches of geometry.
Eduard tech spent only five semesters at Charles University. In 1915 he had to
interrupt his studies and leave for service in the army. After the war he completed his
studies by passing State examinations and for a short period he taught mathematics
at a secondary school in Prague.
In 1920 he received the degree of Doctor of Philosophy from Charles University for
his thesis "On curve and plane elements of the third order". Thereafter tech became
deeply interested in research. He started studying in a systematic manner the differ-
ential projective properties of geometric objects. He became acquainted with papers
by the outstanding Italian geometer G. Fubini and, having obtained a scholarship, he
spent the school year 1921-22 in Turin. Professor Fubini saw the extraordinary ca-
pabilities of young tech and offered him a coauthorship of a monograph. As a result
of the cooperation two volumes of Geometria proiettiva difJerenziale appeared in 1926
and 1927. The authors afterwards wrote another book under the title Introduction a
La geometrie projective difJerentielle des surfaces, published in Paris in 1931.
In 1922, tech submitted a habilitation thesis on projective differential geometry
and became Docent at Charles University. A year later, not yet 30, he was appointed
10 M. Katetov, J. Novak, A. Svec
Extraordinary Professor at the Faculty of Sciences of the Masaryk University in Brno,
where the chair previously held by Matyas Lerch had become vacant. As the chair of
geometry was occupied, he was asked to teach courses in mathematical analysis and
algebra. Therefore he started an intensive study of these disciplines. In a short space
of time he mastered the appropriate literature and for twelve years lectured on analysis
and algebra at this university. This work seems to have had important implications
for his interest in topology.
In 1928 he was appointed Full Professor. At that time he manifested a deep inter-
est in topology. The principal sources he found on the subject were papers published
in Fundamenta mathematicae. He was also influenced by the papers of outstanding
American and Soviet topologists. After 1931 he published no more papers on differen-
tial geometry and devoted himself to research in the field of general and combinatorial
topology. Let us mention the first two papers of pioneer character published in 1932.
One of them is concerned with the general theory of homology in arbitrary spaces and
the other with the general theory of manifolds and theorems of duality; these papers
established Cech's reputation as one of the best specialists in the field of combina-
torial topology. In September 1935 he was invited to a conference on combinatorial
topology held in Moscow. This meeting was attended by a number of the foremost Eu-
ropean and American topologists. Professor Cech reported there on the results of his
research, which met with such attention that he was invited to lecture at the Institute
for Advanced Study in Princeton.
After his return from the U.S.A. in 1936, Cech gave new impulses to the mathemat-
ical research in Brno. With a group of young people deeply interested in mathematics
he founded a topological seminar where at the beginning the papers of P. S. Alexandrov
and P. Urysohn were systematically discussed. The atmosphere of the seminar, as well
as the personality of Professor Cech, who continued to encourage the participants in
their work, had a favourable influence on all its members. Many problems raised by
Cech were solved and, within a period of three years, 26 scientific papers originated in
the seminar. Cech's paper on bicompact spaces was among them. In this paper he in-
vestigated the compactification of completely regular topological spaces now known as
the Cech-Stone compactification. The topological seminar continued till 1939, when af-
ter the German occupation of Bohemia and Moravia all Czech universities were closed.
Nevertheless, even after that Cech met regularly with his closest students, B. Pospisil
and J. Novak, in Pospisil's flat until the arrest of B. Pospisil by the Gestapo in 1941.
Cech's topological seminar holds an important place in the history of Czechoslovak
Life and Work of Eduard Cech 11
mathematics. He introduced there a team-work form of mathematical research.
After twenty-two years of teaching and scientific activity in Brno, Professor Cech
moved to Charles University in Prague in 1945. He became the leading personality in
the organization of Czechoslovak mathematical activities. In 1947 he was appointed the
director of the Mathematical Research Institute of the Czech Academy of Sciences and
Arts. In 1950 the Central Mathematical Institute was established to which Cech was
also appointed director. When the Czechoslovak Academy of Sciences was founded in
1952, this institute was incorporated into the Academy as the Mathematical Institute
of the Czechoslovak Academy of Sciences, again with Cech as its first director. He
laid the foundations of the structure and research orientation of the Institute, aiming
at a balanced development of Czechoslovak mathematics both in the theory and in
the applications in technical as well as biological sciences. In 1954 he returned to
Charles University as a director of the newly founded Mathematical Institute of Charles
University. After 1949 he resumed his own research work and thereafter published
17 papers on differential geometry. Nevertheless, he continued to be interested in
topology; he wrote the book Topologicke prosiory and, just before his death on March
15, 1960, he initiated the first of the Prague Topological Symposia.
The scientific, teaching and organizational activities of Professor Cech contributed
substantially to the development of mathematics in Czechoslovakia. In addition to this
rich involvement, he was deeply interested in the problems of teaching mathematics.
He was one of those mathematicians who understood that there should exist a close
cooperation between university professors and secondary school teachers. Led by this
conviction, he wrote textbooks for secondary schools. In these textbooks he focused
his attention on fixing mathematical concepts in the mind of the pupils and on the
development of abstract logical reasoning.
In a series of pedagogical seminars held in Brno since 1938, Professor Cech devoted
much of his time and energy to the problems of high school mathematics. After 1945
these seminars on elementary mathematics were held both in Prague and Brno.
Professor Cech took part in a number of international mathematical congresses.
He lectured as visiting professor at several European and American universities such
as the University of Warsaw, the University of Lvov, Moscow State University, Prince-
ton University, the University of Michigan, the State University of New York, Harvard
University, to name only some. He was member of the Czechoslovak Academy of Sci-
ences, the Czech Academy of Sciences and Arts, the Royal Czech Society of Sciences,
the Moravian Society of Sciences, an honorary member of the Union of Czechoslo