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**The mathematical legacy of Eduard Čech.**
*(English)*
Zbl 0871.54001

Basel: Birkhäuser Verlag. 445 p. (1993).

From the Foreword: “The work of Profesor Eduard Čech had a significant influence on the development of algebraic and general topology and differential geometry. This book which appears on the occasion of the centenary of Čech’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, Čech-Stone compactification, dimension theory and differential geometry. Each of these includes a selection of Čech’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. Čech’ (1968; Zbl 0167.20501), a list of Čech’s publications and a very brief note on his activity in the didactics of mathematics.”

Contents: M. Katětov, J. Novák and A. Švec: Life and work of Eduard Čech (pp. 9-20); Bibliography of E. Čech (pp. 21-25).

P. Simon: Čech-Stone compactification (pp. 26-37); E. Čech: On bicompact spaces (pp. 38-59); B. Pospíšil: Remark on bicompact spaces (pp. 60-61); I. Gelfand and A. Kolmogoroff: On rings of continuous functions on topological spaces (pp. 62-66); I. Glicksberg: Stone-Čech compactification of products (pp. 67-80); W. Rudin: Homogeneity problems in the theory of Čech compactifications (pp. 81-92); I. I. Parovichenko: On a universal bicompactum of weight \(\aleph\) (pp. 93-96); Z. Frolík: Non-homogeneity of \(\beta P-P\) (pp. 97-99); K. Kunen: Weak \(P\)-points in \(N^*\) (pp. 100-108).

M. Katětov: Dimension theory (pp. 109-129); E. Čech: On the dimension of perfectly normal spaces (pp. 130-148); E. Čech: Contribution to dimension theory (pp. 149-160); O. V. Lokucievskij: On the dimension of bicompacta (pp. 161-164); C. H. Dowker: Inductive dimension of completely normal spaces (pp. 165-177); C. H. Dowker and W. Hurewicz: Dimension of metric spaces (pp. 178-183); P. Vopěnka: On the dimension of compact spaces (pp. 184-190); V. V. Filippov: Bicompacta with distinct dimensions ind and dim (pp. 191-195); 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 (pp. 196-203); M. G. Charalambous: Spaces with noncoinciding dimensions (pp. 204-212).

E. G. Sklyarenko: Algebraic topology (pp. 213-230); E. Čech: General homology theory in an arbitrary space (pp. 231-255); E. Čech: Betti groups of an infinite complex (pp. 256-264); E. Čech: Multiplications on a complex (pp. 265-281); S. Lefschetz: On generalized manifolds (pp. 282-317); C. H. Dowker: Čech cohomology theory and the axioms (pp. 318-332).

I. Kolář: Differential geometry (pp. 333-356); E. Čech: On the surfaces all Segre curves of which are plane curves (pp. 357-392); E. Čech: Developable transformations of line congruences (pp. 393-415); A. Švec: On the differential geometry of a surface embedded in a three dimensional space with projective connection (pp. 416-427); I. Kolář: Order of holonomy of a surface with projective connection (pp. 428-435); B. Cenkl: Geometric deformations of the evolution equations and Bäcklund transformations (pp. 436-438); E. Kraemer: Professor Čech and didactics of mathematics (pp. 439-441).

Contents: M. Katětov, J. Novák and A. Švec: Life and work of Eduard Čech (pp. 9-20); Bibliography of E. Čech (pp. 21-25).

P. Simon: Čech-Stone compactification (pp. 26-37); E. Čech: On bicompact spaces (pp. 38-59); B. Pospíšil: Remark on bicompact spaces (pp. 60-61); I. Gelfand and A. Kolmogoroff: On rings of continuous functions on topological spaces (pp. 62-66); I. Glicksberg: Stone-Čech compactification of products (pp. 67-80); W. Rudin: Homogeneity problems in the theory of Čech compactifications (pp. 81-92); I. I. Parovichenko: On a universal bicompactum of weight \(\aleph\) (pp. 93-96); Z. Frolík: Non-homogeneity of \(\beta P-P\) (pp. 97-99); K. Kunen: Weak \(P\)-points in \(N^*\) (pp. 100-108).

M. Katětov: Dimension theory (pp. 109-129); E. Čech: On the dimension of perfectly normal spaces (pp. 130-148); E. Čech: Contribution to dimension theory (pp. 149-160); O. V. Lokucievskij: On the dimension of bicompacta (pp. 161-164); C. H. Dowker: Inductive dimension of completely normal spaces (pp. 165-177); C. H. Dowker and W. Hurewicz: Dimension of metric spaces (pp. 178-183); P. Vopěnka: On the dimension of compact spaces (pp. 184-190); V. V. Filippov: Bicompacta with distinct dimensions ind and dim (pp. 191-195); 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 (pp. 196-203); M. G. Charalambous: Spaces with noncoinciding dimensions (pp. 204-212).

E. G. Sklyarenko: Algebraic topology (pp. 213-230); E. Čech: General homology theory in an arbitrary space (pp. 231-255); E. Čech: Betti groups of an infinite complex (pp. 256-264); E. Čech: Multiplications on a complex (pp. 265-281); S. Lefschetz: On generalized manifolds (pp. 282-317); C. H. Dowker: Čech cohomology theory and the axioms (pp. 318-332).

I. Kolář: Differential geometry (pp. 333-356); E. Čech: On the surfaces all Segre curves of which are plane curves (pp. 357-392); E. Čech: Developable transformations of line congruences (pp. 393-415); A. Švec: On the differential geometry of a surface embedded in a three dimensional space with projective connection (pp. 416-427); I. Kolář: Order of holonomy of a surface with projective connection (pp. 428-435); B. Cenkl: Geometric deformations of the evolution equations and Bäcklund transformations (pp. 436-438); E. Kraemer: Professor Čech and didactics of mathematics (pp. 439-441).

### MSC:

54-03 | History of general topology |

01A75 | Collected or selected works; reprintings or translations of classics |

00B50 | Collections of translated articles of general interest |

01A70 | Biographies, obituaries, personalia, bibliographies |

55-03 | History of algebraic topology |

54D35 | Extensions of spaces (compactifications, supercompactifications, completions, etc.) |

54F45 | Dimension theory in general topology |

53-03 | History of differential geometry |