We show that there exist -metrizable spaces which do not have the Dugundji extension property ( 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.
@article{bwmeta1.element.bwnjournal-article-fmv150i1p11bwm,
author = {Ian Stares and Jerry Vaughan},
title = {The Dugundji extension property can fail in $\omega$$\mu$ -metrizable spaces},
journal = {Fundamenta Mathematicae},
volume = {149},
year = {1996},
pages = {11-16},
zbl = {0861.54011},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-fmv150i1p11bwm}
}
Stares, Ian; Vaughan, Jerry. The Dugundji extension property can fail in ωµ -metrizable spaces. Fundamenta Mathematicae, Tome 149 (1996) pp. 11-16. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-fmv150i1p11bwm/
[00000] [1] C. J. R. Borges, On stratifiable spaces, Pacific J. Math. 17 (1966), 1-16. | Zbl 0175.19802
[00001] [2] C. J. R. Borges, Absolute extensor spaces: a correction and an answer, Pacific J. Math. 50 (1974), 29-30. | Zbl 0276.54025
[00002] [3] K. Borsuk, Über Isomorphie der Funktionalräume, Bull. Internat. Acad. Polon. Ser. A 1933 (1//3), 1-10. | Zbl 0007.25201
[00003] [4] J. Ceder, Some generalizations of metric spaces, Pacific J. Math. 11 (1961), 105-125. | Zbl 0103.39101
[00004] [5] W. W. Comfort and S. Negrepontis, The Theory of Ultrafilters, Springer, New York, 1974.
[00005] [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. | Zbl 0309.54013
[00006] [7] J. Dugundji, An extension of Tietze's theorem, Pacific J. Math. 1 (1951), 353-367. | Zbl 0043.38105
[00007] [8] R. Engelking, On closed images of the space of irrationals, Proc. Amer. Math. Soc. 21 (1969), 583-586. | Zbl 0177.25501
[00008] [9] R. Engelking, General Topology, Sigma Ser. Pure Math. 6, Heldermann, Berlin, 1989.
[00009] [10] R. W. Heath and D. J. Lutzer, Dugundji extension theorems for linearly ordered spaces, Pacific J. Math. 55 (1974), 419-425. | Zbl 0302.54017
[00010] [11] P. J. Nyikos and H. C. Reichel, Topological characterizations of -metrizable spaces, Topology Appl. 44 (1992), 293-308.
[00011] [12] I. S. Stares, Concerning the Dugundji extension property, Topology Appl. 63 (1995), 165-172.
[00012] [13] J. E. Vaughan, Linearly stratifiable spaces, Pacific J. Math. 43 (1972), 253-265. | Zbl 0226.54027
[00013] [14] S. W. Williams, Box products, in: Handbook of Set-Theoretic Topology, North-Holland, 1984, 169-200.
[00014]