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Proceedings of the American Mathematical Society

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Extension of a result of Dieudonné

Authors: J. M. Worrell and H. H. Wicke
Journal: Proc. Amer. Math. Soc. 25 (1970), 634-637
MSC: Primary 54.30
MathSciNet review: 0264605
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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.

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  • 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
  • D. van Dantzig, Ueber topologisch homogene Kontinua. Fund. Math. 15 (1930), 102-125.
  • 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
  • M. Fréchet, Sur quelques points du calcul fonctionnel, Rend. Circ. Math. Palermo 22 (1906), 1-74.
  • John L. Kelley, General topology, D. Van Nostrand Company, 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
  • J. M. Worrell, Jr., On compact spaces and Čech completeness, Notices Amer. Math. Soc. 13 (1966), 644. Abstract #66T-411.

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Keywords: Complete uniformity, [ill]ech completeness, involutory homogeneity, arc, base of countable order, open mappings, (countably) compact, bicompact, normality, <IMG WIDTH="18" HEIGHT="21" ALIGN="BOTTOM" BORDER="0" SRC="images/img1.gif" ALT="$\aleph$">-wise Lindel&#246;fian, nowhere locally complete, fixed point
Article copyright: © Copyright 1970 American Mathematical Society