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Uniform convergence for a hyperspace


Author: R. J. Gazik
Journal: Proc. Amer. Math. Soc. 42 (1974), 302-306
MSC: Primary 54B20
DOI: https://doi.org/10.1090/S0002-9939-1974-0331304-3
MathSciNet review: 0331304
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Abstract: In this note a uniform convergence in the collection $ C(E)$ of nonempty, compact subsets of a separated uniform convergence space E is defined. This convergence is compared with the hyperspace convergence on $ C(E)$ and it is shown that the two convergences agree on Richardson's class $ \Gamma $. In the case of a regular $ {T_1}$ topological space (E, t) this means that there is a uniform convergence structure on E, which induces t, such that uniform convergence in $ C(E)$ is convergences in the Vietoris topology on $ C(E)$.


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

  • [1] A. C. Cochran, On uniform convergence structures and convergence spaces, Doctoral Dissertation, University of Oklahoma, Norman, Okla., 1966.
  • [2] C. H. Cook and H. R. Fischer, Uniform convergence structures, Math. Ann. 173 (1967), 290-306. MR 36 #845. MR 0217756 (36:845)
  • [3] H. R. Fischer. Limesräume, Math. Ann. 137 (1959), 269-303. MR 22 #225. MR 0109339 (22:225)
  • [4] R. J. Gazik, A hyperspace for convergence spaces, Proc. Amer. Math. Soc. 37 (1973), 234-240. MR 0309042 (46:8153)
  • [5] H. H. Keller, Die Uniformiserbarkeit des Limesräumes, Math. Ann. 176 (1968), 334-341. MR 37 #874. MR 0225280 (37:874)
  • [6] E. A. Michael, Topologies on spaces of subsets, Trans. Amer. Math. Soc. 71 (1951), 152-182. MR 13, 54. MR 0042109 (13:54f)
  • [7] G. D. Richardson, A class of uniform convergence structures, Proc. Amer. Math. Soc. 25 (1970), 399-402. MR 41 #991. MR 0256335 (41:991)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1974-0331304-3
Keywords: Vietoris topology, uniform topology, convergence spaces, uniform convergence spaces, hyperspaces of convergence spaces
Article copyright: © Copyright 1974 American Mathematical Society

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