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Tychonoff's theorem for hyperspaces


Author: Frank A. Chimenti
Journal: Proc. Amer. Math. Soc. 37 (1973), 281-286
MSC: Primary 54B10; Secondary 54B20
DOI: https://doi.org/10.1090/S0002-9939-1973-0307141-1
MathSciNet review: 0307141
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Abstract: If $ \exp ({X_i})\backslash \{ \emptyset \} $ is equipped with a topology that preserves the topological convergence of nets of sets for every $ i \in I$, then the Tychonoff product of the family $ \{ \exp ({X_i})\backslash \{ \emptyset \} :i \in I\} $ is compact if and only if $ {X_i}$ is compact for every $ i \in I$. A similar result concerning sequential compactness is valid, for countable I.


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  • [1] G. Birkhoff, Moore-Smith convergence in general topology, Ann. of Math. (2) 38 (1937), 39-56. MR 1503323
  • [2] D. Bushaw, Elements of general topology, Wiley, New York, 1963, Theorem 14, p. 52. MR 28 #2515. MR 0159298 (28:2515)
  • [3] F. A. Chimenti, On the sequential compactness of the space of subsets, Bull. Polon. Acad. Sci. 20 (1972). MR 0310841 (46:9939)
  • [4] J. M. G. Fell, A Hausdorff topology for the closed subsets of a locally compact non-Hausdorff space, Proc. Amer. Math. Soc. 13 (1962), 472-476. MR 25 #2573. MR 0139135 (25:2573)
  • [5] Z. Frolik, Concerning topological convergence of sets, Czechoslovak Math. J. 10 (85) (1960), 168-180. MR 22 #7098. MR 0116303 (22:7098)
  • [6] F. Hausdorff, Mengenlehre, 3rd ed., de Gruyter, Berlin, 1937; English transl., Chelsea, New York, 1962, p. 168. MR 25 #4999.
  • [7] James Keesling, On the equivalence of normality and compactness in hyperspaces, Pacific J. Math. 33 (1970), 657-667. MR 42 #2418. MR 0267516 (42:2418)
  • [8] Y.-F. Lin, Tychonoff's theorem for the space of multifunctions, Amer. Math. Monthly 74 (1967), 399-400. MR 35 #969. MR 0210074 (35:969)
  • [9] E. Michael, Topologies on spaces of subsets, Trans. Amer. Math. Soc. 71 (1951), 152-182. MR 13, 54. MR 0042109 (13:54f)
  • [10] S. Mrowka, Some comments on the space of subsets, Lecture Notes in Math., vol. 171, Springer-Verlag, Berlin and New York, 1970, pp. 59-63. MR 42 #5216. MR 0270327 (42:5216)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1973-0307141-1
Keywords: Tychonoff product, hyperspace, compact, sequentially compact, Vietoris topology, topological convergence of sets
Article copyright: © Copyright 1973 American Mathematical Society

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