Countable paracompactness, normality, and Moore spaces

Wage, Michael L.

doi:10.1090/S0002-9939-1976-0405364-7

Countable paracompactness, normality, and Moore spaces
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Proc. Amer. Math. Soc. 57 (1976), 183-188 Request permission

Abstract:

In this paper we show that ${\text {MA}} + \neg {\text {CH}}$ implies that there exists a countably paracompact Moore space which is not normal. Further, if there is a model of set theory in which every countably paracompact Moore space is normal, then the normal Moore space conjecture is true in that model. Other examples are given, including a nonnormal space constructed with $\diamond$ which is countably compact, ${T_3}$, first countable, locally compact, perfect, and hereditarily separable.

References

R. H. Bing, Metrization of topological spaces, Canad. J. Math. 3 (1951), 175–186. MR 43449, DOI 10.4153/cjm-1951-022-3
R. H. Bing, Mapping a $3$-sphere onto a homotopy $3$-sphere, Topology Seminar (Wisconsin, 1965) Ann. of Math. Studies, No. 60, Princeton Univ. Press, Princeton, N.J., 1966, pp. 89–99. MR 0219071
Keith J. Devlin and Håvard Johnsbråten, The Souslin problem, Lecture Notes in Mathematics, Vol. 405, Springer-Verlag, Berlin-New York, 1974. MR 0384542, DOI 10.1007/BFb0065979
William G. Fleissner, Separation properties in Moore spaces, Fund. Math. 98 (1978), no. 3, 279–286. MR 478111, DOI 10.4064/fm-98-3-279-286
R. W. Heath, Screenability, pointwise paracompactness, and metrization of Moore spaces, Canadian J. Math. 16 (1964), 763–770. MR 166760, DOI 10.4153/CJM-1964-073-3
A. J. Ostaszewski, On countably compact, perfectly normal spaces, J. London Math. Soc. (2) 14 (1976), no. 3, 505–516. MR 438292, DOI 10.1112/jlms/s2-14.3.505

Examples of non-normal perfect spaces in which countable paracompactness is undetermined

On normality and countable paracompactness

Mary Ellen Rudin, Lectures on set theoretic topology, Conference Board of the Mathematical Sciences Regional Conference Series in Mathematics, No. 23, Published for the Conference Board of the Mathematical Sciences by the American Mathematical Society, Providence, R.I., 1975. Expository lectures from the CBMS Regional Conference held at the University of Wyoming, Laramie, Wyo., August 12–16, 1974. MR 0367886, DOI 10.1090/cbms/023
M. K. Singal and Shashi Prabha Arya, A survey of countably paracompact spaces, Math. Student 40 (1972), 337–372 (1974). MR 397662
Michael L. Wage, A collectionwise Hausdorff, non-normal Moore space, Canadian J. Math. 28 (1976), no. 3, 632–634. MR 405363, DOI 10.4153/CJM-1976-062-5
Phillip Zenor, On countable paracompactness and normality, Prace Mat. 13 (1969), 23–32. MR 0248724