The locally finite topology on
Authors:
G. A. Beer, C. J. Himmelberg, K. Prikry and F. S. Van Vleck
Journal:
Proc. Amer. Math. Soc. 101 (1987), 168172
MSC:
Primary 54B20; Secondary 54A10
MathSciNet review:
897090
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Abstract: Let be a metrizable space. A Vietoristype topology, called the locally finite topology, is defined on the hyperspace of all closed, nonempty subsets of . We show that the locally finite topology coincides with the supremum of all Hausdorff metric topologies corresponding to equivalent metrics on . We also investigate when the locally finite topology coincides with the more usual topologies on and when the locally finite topology is metrizable.
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 [B]
 G. Beer, Metric spaces on which continuous functions are uniformly continuous, Proc. Amer. Math. Soc. 95 (1985), 653658. MR 810180 (87e:54024)
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 E. Klein and A. Thompson, Theory of correspondences, Wiley, New York, 1984. MR 752692 (86a:90012)
 [Ma]
 M. Marjanović, Topologies on collections of closed subsets, Publ. Inst. Math. (Beograd) (N. S.) 20 (1966), 125130. MR 0205214 (34:5047)
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 E. Michael, Topologies on spaces of subsets, Trans. Amer. Math. Soc. 71 (1951), 152182. MR 0042109 (13:54f)
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 J. Nagata, On the uniform topology of bicompactifications, J. Inst. Polytech. Osaka City Univ. 1 (1950), 2838. MR 0037501 (12:272a)
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 J. Rainwater, Spaces whose finest uniformity is metric, Pacific J. Math. 9 (1959), 567570. MR 0106448 (21:5180)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00029939198708970902
PII:
S 00029939(1987)08970902
Keywords:
Hyperspaces,
locally finite topology,
Vietoris topology,
Hausdorff metric topology,
supremum topology,
coincidences,
UC space
Article copyright:
© Copyright 1987
American Mathematical Society
