Does C* -embedding imply C*-embedding in the realm of products with a non-discrete metric factor?

Valentin Gutev; Haruto Ohta

Fundamenta Mathematicae (2000)

  • Volume: 163, Issue: 3, page 241-265
  • ISSN: 0016-2736

Abstract

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The above question was raised by Teodor Przymusiński in May, 1983, in an unpublished manuscript of his. Later on, it was recognized by Takao Hoshina as a question that is of fundamental importance in the theory of rectangular normality. The present paper provides a complete affirmative solution. The technique developed for the purpose allows one to answer also another question of Przymusiński's.

How to cite

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Gutev, Valentin, and Ohta, Haruto. "Does C* -embedding imply C*-embedding in the realm of products with a non-discrete metric factor?." Fundamenta Mathematicae 163.3 (2000): 241-265. <http://eudml.org/doc/212442>.

@article{Gutev2000,
abstract = {The above question was raised by Teodor Przymusiński in May, 1983, in an unpublished manuscript of his. Later on, it was recognized by Takao Hoshina as a question that is of fundamental importance in the theory of rectangular normality. The present paper provides a complete affirmative solution. The technique developed for the purpose allows one to answer also another question of Przymusiński's.},
author = {Gutev, Valentin, Ohta, Haruto},
journal = {Fundamenta Mathematicae},
keywords = {C-embedding; C*-embedding; product space; metric space; product with a metric factor},
language = {eng},
number = {3},
pages = {241-265},
title = {Does C* -embedding imply C*-embedding in the realm of products with a non-discrete metric factor?},
url = {http://eudml.org/doc/212442},
volume = {163},
year = {2000},
}

TY - JOUR
AU - Gutev, Valentin
AU - Ohta, Haruto
TI - Does C* -embedding imply C*-embedding in the realm of products with a non-discrete metric factor?
JO - Fundamenta Mathematicae
PY - 2000
VL - 163
IS - 3
SP - 241
EP - 265
AB - The above question was raised by Teodor Przymusiński in May, 1983, in an unpublished manuscript of his. Later on, it was recognized by Takao Hoshina as a question that is of fundamental importance in the theory of rectangular normality. The present paper provides a complete affirmative solution. The technique developed for the purpose allows one to answer also another question of Przymusiński's.
LA - eng
KW - C-embedding; C*-embedding; product space; metric space; product with a metric factor
UR - http://eudml.org/doc/212442
ER -

References

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  1. [1] R. A. Aló and L. I. Sennott, Extending linear space-valued functions, Math. Ann. 191 (1971), 79-86. Zbl0198.28005
  2. [2] R. A. Aló and L. I. Sennott, Collectionwise normality and the extension of functions on product spaces, Fund. Math. 74 (1972), 231-243. Zbl0245.54009
  3. [3] R. Arens, Extension of coverings, of pseudometrics, and linear-space-valued mappings, Canad. J. Math. 5 (1953), 211-215. 
  4. [4] R. L. Blair and A. W. Hager, Extensions of zero-sets and of real-valued functions, Math. Z. 136 (1974), 41-52. Zbl0264.54011
  5. [5] J. Dugundji, An extension of Tietze's theorem, Pacific J. Math. 1 (1951), 353-367. Zbl0043.38105
  6. [6] T. E. Gantner, Extension of uniformly continuous pseudometrics, Trans. Amer. Math. Soc. 132 (1968), 147-157. Zbl0157.53501
  7. [7] G. Gruenhage, Generalized metric spaces, in: Handbook of Set-Theoretic Topo- logy, K. Kunen and J. E. Vaughan (eds.), North-Holland, Amsterdam, 1984, 423-501. 
  8. [8] T. Hoshina, Spaces with a property related to uniformly local finiteness, Tsukuba J. Math. 6 (1982), 51-62. Zbl0543.54015
  9. [9] T. Hoshina, Extensions of mappings II, in: Topics in General Topology, K. Morita and J. Nagata (eds.), North-Holland, Amsterdam, 1989, 41-80. 
  10. [10] T. Hoshina, Extensions of mappings, in: Recent Progress in General Topology, M. Hušek and J. van Mill (eds.), North-Holland, Amsterdam, 1992, 405-416. Zbl0798.54021
  11. [11] T. Hoshina and K. Yamazaki, C*-embedding and C-embedding on products with certain metric factor, Topology Appl. 82 (1998), 195-204. Zbl0893.54010
  12. [12] J. R. Isbell, Uniform Spaces, Amer. Math. Soc., Providence, RI, 1964. Zbl0124.15601
  13. [13] T. Ishii and H. Ohta, Generalizations of C-embedding and their applications, Math. Japon. 23 (1978), 349-368. Zbl0405.54010
  14. [14] F. Ishikawa, On countably paracompact spaces, Proc. Japan Acad. 31 (1955), 686-687. Zbl0066.41001
  15. [15] M. Katětov, Extensions of locally finite coverings, Colloq. Math. 6 (1958), 145-151. 
  16. [16] K. Kunen, Set Theory. An Introduction to Independence Proofs, Stud. Logic Found. Math. 102, North-Holland, Amsterdam, 1983. 
  17. [17] K. Morita, Paracompactness and product spaces, Fund. Math. 50 (1962), 223-236. Zbl0099.17401
  18. [18] K. Morita, Products of normal spaces with metric spaces, Math. Ann. 154 (1964), 365-382. Zbl0117.39803
  19. [19] K. Morita, On generalizations of Borsuk's homotopy extension theorem, Fund. Math. 88 (1975), 1-6. Zbl0304.55009
  20. [20] K. Morita and T. Hoshina, P-embedding and product spaces, ibid. 93 (1976), 71-80. Zbl0347.54009
  21. [21] H. Ohta, Rectangular normality of products with a metric factor, RIMS Kôkyûroku 823 (1993), 106-117. 
  22. [22] T. C. Przymusiński, Collectionwise normality and extensions of continuous functions, Fund. Math. 98 (1978), 75-81. Zbl0382.54014
  23. [23] T. C. Przymusiński, Extending functions from products with a metric factor and absolutes, Pacific J. Math. 101 (1982), 463-475. Zbl0491.54008
  24. [24] T. C. Przymusiński, Notes on extendability of continuous functions from products with a metric factor, unpublished note, May 1983. 
  25. [25] T. C. Przymusiński, A solution to a problem of E. Michael, Pacific J. Math. 114 (1984), 235-242. 
  26. [26] T. C. Przymusiński and M. Wage, Collectionwise normality and extensions of locally finite coverings, Fund. Math. 109 (1980), 175-187. Zbl0443.54015
  27. [27] M. E. Rudin and M. Starbird, Products with a metric factor, General Topology Appl. 5 (1975), 235-248. 
  28. [28] H. L. Shapiro, Extensions of pseudometrics, Canad. J. Math 18 (1966), 981-998. Zbl0158.41303
  29. [29] M. Starbird, Extending maps from products, in: Studies in Topology, N. H. Stav- rakas and K. R. Allen (eds.), Academic Press, New York, 1975, 559-564. 
  30. [30] K. Yamazaki, C*-embedding and C-embedding on product spaces, Tsukuba J. Math. 21 (1997), 515-527. Zbl0887.54017

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