A normal first countable ccc nonseparable space

Bell, Murray G.

doi:10.1090/S0002-9939-1979-0521889-0

A normal first countable ccc nonseparable space

Author: Murray G. Bell
Journal: Proc. Amer. Math. Soc. 74 (1979), 151-155
MSC: Primary 54D15; Secondary 54G20
DOI: https://doi.org/10.1090/S0002-9939-1979-0521889-0
MathSciNet review: 521889
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Abstract: We construct an absolute example of a space having the properties in the title. Let Y be the set of nonempty finite subsets of the Cantor cube of countable weight. The Pixley-Roy topology on Y is not normal, but the Vietoris topology on Y is normal. Our space can be considered a normalization of the Pixley-Roy topology on Y by adding cluster points which as a subspace have the Vietoris topology. The Alexandroff duplicating procedure is used liberally to glue the space together. The example is also a sigma compact paracompact p-space.

If further set-theoretic assumptions are made (e.g. or ${\text{MA}} + \neg {\text{CH}}$ ), then it is known that even perfectly normal such examples exist.

[1] A. V. Arhangel' skiĭ, On a class of spaces containing all metric spaces and all locally bicompact spaces, Amer. Math. Soc. Transl. (2) 92 (1970), 1-39.
[2] Murray Bell, John Ginsburg, and Grant Woods, Cardinal inequalities for topological spaces involving the weak Lindelof number, Pacific J. Math. 79 (1978), no. 1, 37–45. MR 526665
[3] Eric K. van Douwen, The Pixley-Roy topology on spaces of subsets, Set-theoretic topology (Papers, Inst. Medicine and Math., Ohio Univ., Athens, Ohio, 1975-1976) Academic Press, New York, 1977, pp. 111–134. MR 0440489
[4] George M. Reed (ed.), Set-theoretic topology, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1977. Papers presented at a series of Conferences, Seminars, and Colloquia, held at the Institute for Medicine and Mathematics, Ohio Univ., Athens, Ohio, 1975–1976. MR 0431077
[5] E. van Douwen and T. C. Przymusiński, First countable and countable spaces all compactifications of which contain $\beta N$ , Fund. Math. (to appear).
[6] A. Hajnal and I. Juhász, A consequence of Martin’s axiom, Nederl. Akad. Wetensch. Proc. Ser. A 74=Indag. Math. 33 (1971), 457–463. MR 0305344
[7] Tomáš Jech, Non-provability of Souslin’s hypothesis, Comment. Math. Univ. Carolinae 8 (1967), 291–305. MR 0215729
[8] Ernest Michael, Topologies on spaces of subsets, Trans. Amer. Math. Soc. 71 (1951), 152–182. MR 0042109, https://doi.org/10.1090/S0002-9947-1951-0042109-4
[9] Carl Pixley and Prabir Roy, Uncompletable Moore spaces, Proceedings of the Auburn Topology Conference (Auburn Univ., Auburn, Ala., 1969; dedicated to F. Burton Jones on the occasion of his 60th birthday), Auburn Univ., Auburn, Ala., 1969, pp. 75–85. MR 0397671
[10] T. Przymusiński and F. D. Tall, The undecidability of the existence of a non-separable normal Moore space satisfying the countable chain condition, Fund. Math. 85 (1974), 291–297. MR 0367934, https://doi.org/10.4064/fm-85-3-291-297
[11] Mary Ellen Rudin, Lectures on set theoretic topology, 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; Conference Board of the Mathematical Sciences Regional Conference Series in Mathematics, No. 23. MR 0367886
[12] R. M. Solovay and S. Tennenbaum, Iterated Cohen extensions and Souslin’s problem, Ann. of Math. (2) 94 (1971), 201–245. MR 0294139, https://doi.org/10.2307/1970860
[13] Franklin D. Tall, On the existence of non-metrizable hereditarily Lindelöf spaces with point-countable bases, Duke Math. J. 41 (1974), 299–304. MR 0388339
[14] V. M. Ul′janov, Examples of finally compact spaces that do not have bicompact extensions of countable character, Dokl. Akad. Nauk SSSR 220 (1975), 1282–1285 (Russian). MR 0405378
[15] Leopold Vietoris, Kontinua zweiter Ordnung, Monatsh. Math. Phys. 33 (1923), no. 1, 49–62 (German). MR 1549268, https://doi.org/10.1007/BF01705590
[16] Stephen Willard, General topology, Addison-Wesley Publishing Co., Reading, Mass.-London-Don Mills, Ont., 1970. MR 0264581