Extension of a result of Dieudonné

Worrell, J. M.; Wicke, H. H.

doi:10.1090/S0002-9939-1970-0264605-4

Extension of a result of Dieudonné
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by J. M. Worrell and H. H. Wicke

Proc. Amer. Math. Soc. 25 (1970), 634-637

DOI: https://doi.org/10.1090/S0002-9939-1970-0264605-4

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Abstract:

Dieudonné showed that there exists a normal (countably) compact uniform ${T_1}$-space which has no topology preserving complete uniformity [4]. His example, being the space of the countable ordinals with respect to the order topology, everywhere locally has a complete uniformity. Here we show, as a corollary to Dieudonné’s result and a result of Worrell [10], that there exists a normal (countably) compact first countable involutorily homogeneous uniform ${T_1}$-space locally homeomorphic with itself which has no topology preserving complete uniformity.

References

A. V. Arhangel′skiĭ, Some metrization theorems, Uspehi Mat. Nauk 18 (1963), no. 5 (113), 139–145 (Russian). MR 0156318
R. H. Bing, Metrization of topological spaces, Canad. J. Math. 3 (1951), 175–186. MR 43449, DOI 10.4153/cjm-1951-022-3

Ueber topologisch homogene Kontinua

15

Jean Dieudonné, Un exemple d’espace normal non susceptible d’une structure uniforme d’espace complet, C. R. Acad. Sci. Paris 209 (1939), 145–147 (French). MR 175

Sur quelques points du calcul fonctionnel

22

John L. Kelley, General topology, D. Van Nostrand Co., Inc., Toronto-New York-London, 1955. MR 0070144
R. L. Moore, Foundations of point set theory, Revised edition, American Mathematical Society Colloquium Publications, Vol. XIII, American Mathematical Society, Providence, R.I., 1962. MR 0150722
Howard H. Wicke, The regular open continuous images of complete metric spaces, Pacific J. Math. 23 (1967), 621–625. MR 219035
H. H. Wicke and J. M. Worrell Jr., Open continuous mappings of spaces having bases of countable order, Duke Math. J. 34 (1967), 255–271. MR 210084

On compact spaces and Čech completeness

13