The Dugundji extension property can fail in ωµ -metrizable spaces

Ian Stares; Jerry Vaughan

Fundamenta Mathematicae (1996)

  • Volume: 150, Issue: 1, page 11-16
  • ISSN: 0016-2736

Abstract

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We show that there exist ω μ -metrizable spaces which do not have the Dugundji extension property ( 2 ω 1 with the countable box topology is such a space). This answers a question posed by the second author in 1972, and shows that certain results of van Douwen and Borges are false.

How to cite

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Stares, Ian, and Vaughan, Jerry. "The Dugundji extension property can fail in ωµ -metrizable spaces." Fundamenta Mathematicae 150.1 (1996): 11-16. <http://eudml.org/doc/212158>.

@article{Stares1996,
abstract = {We show that there exist $ω_μ$-metrizable spaces which do not have the Dugundji extension property ($2^\{ω_1\}$ with the countable box topology is such a space). This answers a question posed by the second author in 1972, and shows that certain results of van Douwen and Borges are false.},
author = {Stares, Ian, Vaughan, Jerry},
journal = {Fundamenta Mathematicae},
keywords = {Dugundji extension theorem; $ω_μ$-metrizable spaces; box topology; Baire category; Michael line; -metrizable spaces; unbounded monotone extension; linearly stratifiable spaces},
language = {eng},
number = {1},
pages = {11-16},
title = {The Dugundji extension property can fail in ωµ -metrizable spaces},
url = {http://eudml.org/doc/212158},
volume = {150},
year = {1996},
}

TY - JOUR
AU - Stares, Ian
AU - Vaughan, Jerry
TI - The Dugundji extension property can fail in ωµ -metrizable spaces
JO - Fundamenta Mathematicae
PY - 1996
VL - 150
IS - 1
SP - 11
EP - 16
AB - We show that there exist $ω_μ$-metrizable spaces which do not have the Dugundji extension property ($2^{ω_1}$ with the countable box topology is such a space). This answers a question posed by the second author in 1972, and shows that certain results of van Douwen and Borges are false.
LA - eng
KW - Dugundji extension theorem; $ω_μ$-metrizable spaces; box topology; Baire category; Michael line; -metrizable spaces; unbounded monotone extension; linearly stratifiable spaces
UR - http://eudml.org/doc/212158
ER -

References

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  1. [1] C. J. R. Borges, On stratifiable spaces, Pacific J. Math. 17 (1966), 1-16. Zbl0175.19802
  2. [2] C. J. R. Borges, Absolute extensor spaces: a correction and an answer, Pacific J. Math. 50 (1974), 29-30. Zbl0276.54025
  3. [3] K. Borsuk, Über Isomorphie der Funktionalräume, Bull. Internat. Acad. Polon. Ser. A 1933 (1//3), 1-10. Zbl0007.25201
  4. [4] J. Ceder, Some generalizations of metric spaces, Pacific J. Math. 11 (1961), 105-125. Zbl0103.39101
  5. [5] W. W. Comfort and S. Negrepontis, The Theory of Ultrafilters, Springer, New York, 1974. 
  6. [6] E. K. van Douwen, Simultaneous extension of continuous functions, in: E. K. van Douwen, Collected Papers, Vol. 1, J. van Mill (ed.), North-Holland, Amsterdam, 1994. Zbl0309.54013
  7. [7] J. Dugundji, An extension of Tietze's theorem, Pacific J. Math. 1 (1951), 353-367. Zbl0043.38105
  8. [8] R. Engelking, On closed images of the space of irrationals, Proc. Amer. Math. Soc. 21 (1969), 583-586. Zbl0177.25501
  9. [9] R. Engelking, General Topology, Sigma Ser. Pure Math. 6, Heldermann, Berlin, 1989. 
  10. [10] R. W. Heath and D. J. Lutzer, Dugundji extension theorems for linearly ordered spaces, Pacific J. Math. 55 (1974), 419-425. Zbl0302.54017
  11. [11] P. J. Nyikos and H. C. Reichel, Topological characterizations of ω μ -metrizable spaces, Topology Appl. 44 (1992), 293-308. 
  12. [12] I. S. Stares, Concerning the Dugundji extension property, Topology Appl. 63 (1995), 165-172. 
  13. [13] J. E. Vaughan, Linearly stratifiable spaces, Pacific J. Math. 43 (1972), 253-265. Zbl0226.54027
  14. [14] S. W. Williams, Box products, in: Handbook of Set-Theoretic Topology, North-Holland, 1984, 169-200. 
  15. </REFERENCES> 

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