Ordinal products of topological spaces

Vitalij Chatyrko

Fundamenta Mathematicae (1994)

  • Volume: 144, Issue: 2, page 95-117
  • ISSN: 0016-2736

Abstract

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The notion of the ordinal product of a transfinite sequence of topological spaces which is an extension of the finite product operation is introduced. The dimensions of finite and infinite ordinal products are estimated. In particular, the dimensions of ordinary products of Smirnov's [S] and Henderson's [He1] compacta are calculated.

How to cite

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Chatyrko, Vitalij. "Ordinal products of topological spaces." Fundamenta Mathematicae 144.2 (1994): 95-117. <http://eudml.org/doc/212023>.

@article{Chatyrko1994,
abstract = {The notion of the ordinal product of a transfinite sequence of topological spaces which is an extension of the finite product operation is introduced. The dimensions of finite and infinite ordinal products are estimated. In particular, the dimensions of ordinary products of Smirnov's [S] and Henderson's [He1] compacta are calculated.},
author = {Chatyrko, Vitalij},
journal = {Fundamenta Mathematicae},
keywords = {ordinal product; sequences of spaces; dimension functions; transfinite dimension; product},
language = {eng},
number = {2},
pages = {95-117},
title = {Ordinal products of topological spaces},
url = {http://eudml.org/doc/212023},
volume = {144},
year = {1994},
}

TY - JOUR
AU - Chatyrko, Vitalij
TI - Ordinal products of topological spaces
JO - Fundamenta Mathematicae
PY - 1994
VL - 144
IS - 2
SP - 95
EP - 117
AB - The notion of the ordinal product of a transfinite sequence of topological spaces which is an extension of the finite product operation is introduced. The dimensions of finite and infinite ordinal products are estimated. In particular, the dimensions of ordinary products of Smirnov's [S] and Henderson's [He1] compacta are calculated.
LA - eng
KW - ordinal product; sequences of spaces; dimension functions; transfinite dimension; product
UR - http://eudml.org/doc/212023
ER -

References

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  1. [A-Pa] P. S. Aleksandrov and B. A. Pasynkov, Introduction to Dimension Theory, Nauka, Moscow, 1973 (in Russian). 
  2. [B1] P. Borst, Classification of weakly infinite-dimensional spaces. Part I: A transfinite extension of the covering dimension, Fund. Math. 130 (1988), 1-25. Zbl0661.54035
  3. [B2] P. Borst, Classification of weakly infinite-dimensional spaces. Part II: Essential mappings, ibid., 73-99. Zbl0661.54036
  4. [B3] P. Borst, Some remarks concerning C-spaces, preprint. 
  5. [D] A. N. Dranishnikov, Absolute extensors in dimension n and dimension raising n-soft mappings, Uspekhi Mat. Nauk 39 (5) (1984), 55-95 (in Russian). 
  6. [E1] R. Engelking, General Topology, PWN, Warszawa 1977. 
  7. [E2] R. Engelking, Dimension Theory, PWN, Warszawa 1978. 
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  10. [Ha] Y. Hattori, Solution of problems concerning transfinite dimension, Questions Answers Gen. Topology 1 (1983), 128-134. 
  11. [Ha-Y] Y. Hattori and K. Yamada, Closed pre-images of C-spaces, Math. Japon. 34 (1989), 555-561. Zbl0694.54028
  12. [H] F. Hausdorff, Set Theory, Chelsea, New York, 1962. 
  13. [He1] D. W. Henderson, A lower bound for transfinite dimension, Fund. Math. 63 (1968), 167-173. Zbl0167.51301
  14. [He2] D. W. Henderson, D-dimension I. A new transfinite dimension, Pacific J. Math. 26 (1968), 91-107. Zbl0162.26904
  15. [Hes] G. Hessenberg, Grundbegriffe der Mengenlehre, Göttingen, 1906. Zbl37.0067.03
  16. [K-M] K. Kuratowski and A. Mostowski, Set Theory, PWN and North-Holland, 1976. 
  17. [Le] B. T. Levshenko, Spaces of transfinite dimensionality, Mat. Sb. 67 (1965), 255-266 (in Russian); English transl.: Amer. Math. Soc. Transl. (2) 73 (1968), 135-148. Zbl0193.51305
  18. [L] L. A. Luxemburg, On compacta with non-coinciding transfinite dimensions, Dokl. Akad. Nauk SSSR 212 (1973), 1297-1300 (in Russian); English transl.: Soviet Math. Dokl. 14 (1973), 1593-1597. 
  19. [Pa1] B. A. Pasynkov, On dimension of rectangular products, Dokl. Akad. Nauk SSSR 221 (1975), 291-294 (in Russian). 
  20. [Pa2] B. A. Pasynkov, On transfinite dimension, Abstracts of Leningrad Internat. Topology Conf., 1982 (in Russian). 
  21. [P] R. Pol, On classification of weakly infinite-dimensional compacta, Fund. Math. 116 (1983), 169-188. Zbl0571.54030
  22. [Po] L. Polkowski, On transfinite dimension, Colloq. Math. 50 (1985), 61-79. Zbl0613.54024
  23. [S] Yu. M. Smirnov, On universal spaces for some classes of infinite-dimensional spaces, Izv. Akad. Nauk SSSR 23 (1959), 185-196 (in Russian); English transl.: Amer. Math. Soc. Transl. (2) 21 (1962), 21-34. 
  24. [T] G. H. Toulmin, Shuffling ordinals and transfinite dimension, Proc. London Math. Soc. 4 (1954), 177-195. Zbl0055.41406

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