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Transactions of the American Mathematical Society

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Supercompactness of compactifications and hyperspaces


Author: Murray G. Bell
Journal: Trans. Amer. Math. Soc. 281 (1984), 717-724
MSC: Primary 54D30; Secondary 54B20
MathSciNet review: 722770
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Abstract: We prove a theorem which implies that if $ \gamma \omega $ is a supercompact compactification of the countable discrete space $ \omega $ then $ \gamma \omega - \omega $ is separable. This improves an earlier result of the author's that such a $ \gamma \omega $ must have $ \gamma \omega - \omega \;{\text{ccc}}$. We prove a theorem which implies that the hyperspace of closed subsets of $ {2^{\omega_2}}$ is not a continuous image of a supercompact space. This improves an earlier result of $ {\text{L}}$. Šapiro that the hyperspace of closed subsets of $ {2^{\omega_2}}$ is not dyadic.


References [Enhancements On Off] (What's this?)

  • [1] P. Alexandroff, Zur Theorie der topologischen Räume, C. R. (Doklady) Acad. Sci. URSS 11 (1936), 55-58.
  • [2] Murray G. Bell, A cellular constraint in supercompact Hausdorff spaces, Canad. J. Math. 30 (1978), no. 6, 1144–1151. MR 511552, 10.4153/CJM-1978-095-7
  • [3] Murray G. Bell, Compact ccc nonseparable spaces of small weight, The Proceedings of the 1980 Topology Conference (Univ. Alabama, Birmingham, Ala., 1980), 1980, pp. 11–25 (1981). MR 624458
  • [4] -, Two Boolean algebras with extreme cellular and compactness properties, preprint.
  • [5] Murray G. Bell and Jan van Mill, The compactness number of a compact topological space. I, Fund. Math. 106 (1980), no. 3, 163–173. MR 584490
  • [6] Eric K. van Douwen, Mappings from hyperspaces and convergent sequences, Topology Appl. 34 (1990), no. 1, 35–45. MR 1035458, 10.1016/0166-8641(90)90087-I
  • [7] Eric van Douwen and Jan van Mill, Supercompact spaces, Topology Appl. 13 (1982), no. 1, 21–32. MR 637424, 10.1016/0166-8641(82)90004-9
  • [8] B. Efimov, The imbedding of the Stone-Čech compactifications of discrete spaces into bicompacta, Dokl. Akad. Nauk SSSR 189 (1969), 244–246 (Russian). MR 0253290
  • [9] Contributions to extension theory of topological structures, Proceedings of the Symposium held in Berlin, August 14–19, vol. 1967, VEB Deutscher Verlag der Wissenschaften, Berlin, 1969. MR 0244955
  • [10] I. Juhász, Cardinal functions in topology--ten years later, Math. Centre Tracts, No. 123, Mathematisch Centrum, Amsterdam, 1980.
  • [11] J. van Mill, Supercompactness and Wallman spaces, Mathematisch Centrum, Amsterdam, 1977. Mathematical Centre Tracts, No. 85. MR 0464160
  • [12] Charles F. Mills and Jan van Mill, A nonsupercompact continuous image of a supercompact space, Houston J. Math. 5 (1979), no. 2, 241–247. MR 546758
  • [13] S. Mrówka, Mazur theorem and 𝑚-adic spaces, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 18 (1970), 299–305 (English, with Loose Russian summary). MR 0264613
  • [14] L. B. S $ {D^{\aleph_2}}$ is not a dyadic bicompact, Soviet Math. Dokl. 17 (1976), 937-941.
  • [15] S. Sirota, Spectral representation of spaces of closed subsets of bicompacta, Soviet Math. Dokl. 9 (1968), 997-1000.
  • [16] Leopold Vietoris, Bereiche zweiter Ordnung, Monatsh. Math. Phys. 32 (1922), no. 1, 258–280 (German). MR 1549179, 10.1007/BF01696886

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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1984-0722770-0
Keywords: Supercompact, hyperspace, dyadic
Article copyright: © Copyright 1984 American Mathematical Society